---

Hyperbolicity is Dense in the Real Quadratic Family

arXiv:math/9207219v1 [math.DS] 6 Jul 1992Hyperbolicity is Dense in the Real Quadratic FamilyGrzegorz ´Swi¸atekMathematics DepartmentPrinceton UniversityPrinceton, NJ 08544, USA ∗March 5, 1994Stony Brook IMS Preprint #1992/10July 1992revised March 1994AbstractIt is shown that for non-hyperbolic real quadratic polynomials topological and qua-sisymmetric conjugacy classes are the same.By quasiconformal rigidity, each class has only one representative in the quadraticfamily, which proves that hyperbolic maps are dense.1Fundamental concepts1.1IntroductionStatement of the results.Dense Hyperbolicity Theorem In the real quadratic familyfa(x) = ax(1 −x) , 0 < a ≤4the mapping fa has an attracting cycle, and thus is hyperbolic on its Julia set, for an openand dense set of parameters a.What we actually prove is this:Main TheoremLet f and ˆf be two real quadratic polynomials with a bounded forwardcritical orbit and no attracting or indifferent cycles. Then, if they are topologically conjugate,the conjugacy extends to a quasiconformal conjugacy between their analytic continuations tothe complex plane.∗The author gratefully acknowledges partial support from NSF Grant #DMS-9206793 and the SloanFoundation.

Derivation of the Dense Hyperbolicity Theorem.We show that the Main The-orem implies the Dense Hyperbolicity Theorem. Quasiconformal conjugacy classes of nor-malized complex quadratic polynomials are known to be either points or open (see [21].

)We remind the reader, see [25], that the kneading sequence is aperiodic for a real quadraticpolynomial precisely when this polynomial has no attracting or indifferent periodic orbits.Therefore, by the Main Theorem, topological conjugacy classes of real quadratic polynomialswith aperiodic kneading sequences are either points or open in the space of real parametersa. On the other hand, it is an elementary observation that the set of polynomials with thesame aperiodic kneading sequence in the real quadratic family is also closed.

So, for everyaperiodic kneading sequence there is at most one polynomial in the real quadratic familywith this kneading sequence.Next, between two parameter values a1 and a2 for which different kneading sequencesoccur, there is a parameter a so that fa has a periodic kneading sequence. So, the onlyway the Dense Hyperbolicity Theorem could fail is if there were an interval filled withpolynomials without attracting periodic orbits and yet with periodic kneading sequences.Such polynomials would all have to be parabolic (have indifferent periodic orbits).

It well-known, however, by the work of [6], that there are only countably many such polynomials.The Dense Hyperbolicity Theorem follows.Consequences of the theorems.The Dense Hyperbolicity Conjecture had a longhistory. In a paper from 1920, see [7], Fatou expressed the belief that “general” (generic intoday’s language?) rational maps are expanding on the Julia set.

Our result may be re-garded as progress in the verification of his conjecture. More recently, the fundamental workof Milnor and Thurston, see [25], showed the monotonicity of the kneading invariant in thequadratic family.

They also conjectured that the set of parameter values for which attractiveperiodic orbits exist is dense, which means that the kneading sequence is strictly increas-ing unless it is periodic. The Dense Hyperbolicity Theorem implies Milnor and Thurston’sconjecture.

Otherwise, we would have an interval in the parameter space filled with polyno-mials with an aperiodic kneading sequence, in a clear violation of the Dense HyperbolicityTheorem.The Main Theorem and other results.Yoccoz, [28], proved that a non-hyperbolicquadratic polynomial with a fixed non-periodic kneading sequence is unique up to an affineconjugacy unless it is infinitely renormalizable.Thus, we only need to prove our MainTheorem if the maps are infinitely renormalizable. However, our approach automaticallygives a proof for all non-hyperbolic polynomials, so we provide an independent argument.The work of [27] proved the Main Theorem for infinitely renormalizable polynomials ofbounded combinatorial type.

The paper [18] proved the Main Theorem for some infinitelyrenormalizable quadratic polynomials not covered by [27]. A different approach to Fatou’sconjecture was taken in a recent paper [22].

That work proves that there is no invariant linefield on the Julia set of an infinitely renormalizable real polynomial. This result implies thatthere is no non-hyperbolic component of the Mandelbrot set containing this real polynomialin its interior, however it is not known if it can also imply the Dense Hyperbolicity Theorem.2

Beginning of the proof.Our method is based on the direct construction of a quasicon-formal conjugacy and relies on techniques developed in [17] and [18].The pull-back construction shown in [27] allows one to pass from quasisymmetric conju-gacy classes on the real line to quasiconformal conjugacies in the complex plane. Thus, theMain Theorem is reduced to conjugacies on the real line.We can reduce the proof of the Main Theorem, to the following Reduced Theorem:Reduced Theorem Let f and ˆf be two real quadratic polynomials with the same aperiodickneading sequence and bounded forward critical orbits.

Normalize them to the form x →ax(1 −x). Then the conjugacy between f and ˆf on the interval [0, 1] is quasisymmetric.This paper relies heavily on [17], which describes the inducing process on the real line,and [18] which worked out many ideas and estimates that we use.Questions remaining.Does the Main Theorem remain true for quadratic S-unimodalmaps?

I believe that the difficulties here are only technical in nature and the answer shouldbe affirmative. However, in that case the Dense Hyperbolicity Theorem will not follow fromthe Main Theorem.Is the Main Theorem true for unimodal polynomials of higher even degree?

The presentproof uses degree 2 in the proof of Theorem 4. Theorem 4 seems an irreplaceable element ofthe proof.Our result implies that the polynomials for which a homoclinic tangency occurs (the criti-cal orbit meets a repelling periodic orbit) are dense in the set of non-hyperbolic polynomials.On the other hand, in view of [15], those correspond to density points in the parameterspace of the set of polynomials with an absolutely continuous invariant measure.

It seemsreasonable to conjecture that the set of non-hyperbolic maps without an absolutely contin-uous invariant measure has 0 Lebesgue measure. This problem was posed by J. Palis duringthe workshop in Trieste in June 1992.

We do not know of any recent progress in solving thisproblem.Acknowledgements.Many ideas of the proof come from [17] and [18] which were donejointly with Michael Jakobson. His support in the preparation of the present paper wasalso crucial.

Another important source of my ideas were Dennis Sullivan’s lectures which Iheard in New York in 1988. I am also grateful to Jean-Christophe Yoccoz for pointing outto certain deficiencies of the original draft.

Jacek Graczyk helped me with many discussionsas well as by giving the idea of the proof of Proposition 2.1.2Outline of the paper.In order to prove the Reduced Theorem we apply the inducing construction, essentiallysimilar to the one used in [18], to f and ˆf. We also develop the technique for constructingquasiconformal “branchwise equivalences” in a parallel pull-back construction.

The infinitelyrenormalizable case is treated by constructing a “saturated map” on each stage of renormal-ization, together with a uniformly quasisymmetric branchwise equivalence, and sewing themto get the quasisymmetric conjugacy. These are the same ideas as used in [18].3

The rest of section 1 is devoted to defining and introducing main concepts of the proof.We also reduce the Reduced Theorem to an even simpler Theorem 1. Theorem 1 allowsone to eliminate renormalization from the picture and proceed in almost the same way inrenormalizable and non-renormalizable cases.The results of section 2 are summarized in Theorem 2.

Theorem 2 represents the be-ginning stage in the construction of induced mappings and branchwise equivalences. Ourmain technique of “complex pull-back”, introduced later in section 3, may not immediatelyapply to high renormalizations of a polynomial, since those are not known to be complexpolynomial-like in the sense of [6].

For this reason we are forced to proceed mostly by realmethods introduced in [18]. This section also contains an important new lemma about nearlyparabolic S-unimodal mappings, i.e.

Proposition 2.In section 3 we introduce our powerful tool for constructing quasiconformal branchwiseequivalences. This combines certain ideas of citekus (internal marking) with complex pull-back similar to a construction used in [5].

The main features of the construction are describedby Theorem 3. We then proceed to prove Theorem 4.

Theorem 4 describes the conformalgeometry of our so-called “box case”, which is somewhat similar to the persistently recurrentcase studied by [28]. Another proof of Theorem 4 can be found in [10].

However, the proofwe give is simpler once we can apply our technique of complex pull-back of branchwiseequivalences.In section 4 we apply the complex pull-back construction to the induced objects obtainedby Theorem 2. Estimates are based on Theorems 3 and 4.

The results of this section aregiven by Theorem 5.Section 5 concludes the proof of Theorem 1 from Theorem 5. The construction of satu-rated mappings follows the work of [18] quite closely.The Appendix contains a result related to Theorem 4 and illustrates the technique ofseparating annuli on which the work of [10], referenced from this paper, is based.

The resultof the Appendix is not, however, an integral part of the proof of our main theorems.To help the reader (and the author as well) to cope with the size of the paper, we tried tomake all sections, with the exception of section 1, as independent as possible. Cross-sectionreferences are mostly limited to the Theorems so that, hopefully, each section can be studiedindependently.1.3Induced mappingsWe define a class of unimodal mappings.Definition 1.1 For η > 0, we define the class Fη to comprise all unimodal mappings of theinterval [0, 1] into itself normalized so that 0 is a fixed point which satisfy these conditions:• Any f ∈F can be written as h(x2) where h is a polynomial defined on a set containing[0, 1] with range (−1 −η, 1 + η).• The map h has no critical values except on the real line.• The Schwarzian derivative of h is non-positive.4

• The mapping f has no attracting or indifferent periodic cycles.• The critical orbit is recurrent.We also defineF :=[η>0Fη .We observe that class F contains all infinitely renormalizable quadratic polynomials andtheir renormalizations, up to an affine change of coordinates.Induced mapsThe method of inducing was applied to the study of unimodal maps firstin [15], then in [13]. In [18] and [17] an elaborate approach was developed to study inducedmaps, that is, transformations defined to be iterations of the original unimodal map restrictedto pieces of the domain.

We define a more general and abstract notion in this work, namely:Definition 1.2 A generalized induced map φ on an interval J is assumed to satisfy thefollowing conditions:• the domain of φ, called U, is an open and dense subset of J,• φ maps into J,• restricted to each connected component φ is a polynomial with all critical values on thereal line and with negative Schwarzian derivative,• all critical points of φ are of order 2 and each connected component of U contains atmost one critical point of φ.A restriction of a generalized induced map to a connected component of its domain willbe called a branch of φ. Depending on whether the domain of this branch contains thecritical point or not, the branch will be called folding or monotone.

Domains of branches ofφ will also be referred to as domains of φ, not to be confused with the domain of φ which isU. In most cases generalized induced maps should be thought of as piecewise iterations of amapping from F. If they do not arise in this way, we will describe them as artificial maps.The fundamental inducing domain.By the assumption that all periodic orbits arerepelling, every f ∈F has a fixed point q > 0.Definition 1.3 If f ∈F, we define the fundamental inducing domain of f. Consider thefirst return time of the critical point to the interval (−q, q).

If it is not equal to 3, or it isequal to 3 and there is a periodic point of period 3, then the fundamental inducing domainis (−q, q). Otherwise, there is a periodic point q′ < 0 of period 2 inside (−q, q).

Then, thefundamental inducing domain is (q′, −q′).5

Branchwise equivalences.Definition 1.4 Given two generalized induced mappings on J and ˆJ respectively, a branch-wise equivalence between them is an orientation preserving homeomorphism of J onto ˆJwhich maps the domain U of the first map onto the domain ˆU of the second map.So the notion of a branchwise equivalence is independent of the dynamics, only of domainsof the generalized induced mappings.1.4Conjugacy between renormalizable mapsThe Real K¨obe Lemma.Consider a diffeomorphism h onto its image (b, c). Suppose thatits has an extension ˜h onto a larger image (a, d) which is still a diffeomorphism.

Provided that˜h has negative Schwarzian derivative, and |a−b|·|c−d||c−a|·|d−b| ≥ǫ, we will say that h is ǫ-extendible.The following holds for ǫ-extendible maps:Fact 1.1 There is a function C of ǫ only so that C(ǫ) →0 as ǫ →1 and for every h definedon an interval I and ǫ-extendible,|N h| · |I| ≤C(ǫ) .Proof:Apart from the limit behavior as ǫ goes to 1, this fact is proved in [23], Theorem IV.1.2 .The asymptotic behavior can be obtained from Lemma 1 of [11] which says that if ˜h mapsthe unit interval into itself, thenN h(x) ≤2h′(x)dist ({0, 1}, h(x)) . (1)The normalization condition can be satisfied by pre- and post-composing ˜h with affine maps.This will not change N ˜h·|I|, so we just assume that ˜h is normalized.

Since we are interestedin ǫ close to 1, the denominator of (1) is large and h′(x) is no more thanexp C(12)|h(I)||I|.As |h(I)| goes to 0 with ǫ growing to 1, we are done.Q.E.D.Properties of renormalization.A mapping f ∈F will be called renormalizable providedthat a restrictive interval exists for f. An open interval J symmetric with respect to 0 will becalled restrictive if for some n > 1 intervals J, f(J), f n(J) are disjoint, whereas f n(J) ⊂J.These definitions are broadly used in literature and can be traced back at least to [12]. Givenan f, the notions of a locally maximal and maximal restrictive interval will be used whichare self-explanatory.

Observe that if J is locally maximal, then f ∂J ⊂∂J.6

If f is renormalizable, J is its maximal restrictive interval and n is the first returntime form J into itself, we can consider f n restricted to J which will be called the firstrenormalization of f.If f is in F, we define its renormalization sequence f0, f1, · · · , fω.Here ω can be finite or infinity meaning that the sequence is infinite.The definition isinductive. f0 is f. If fi is renormalizable, then fi+1 is the first renormalization of fi.

Iffi is non-renormalizable, the sequence ends.The original mapping f is called infinitelyrenormalizable if ω = ∞, finitely renormalizable if 0 < ω < ∞and non-renormalizable ifω = 0.Distortion in renormalization sequences.Fact 1.2 1.1 Let f ∈Fη and fi be the renormalization sequence. For every η > 0 there is a˜η > 0 so that for every i fi belongs to F˜η after an affine change of coordinates.Proof:A similar estimate appeared in [27].

Our version appears as a step in the proof of LemmaVI.2.1 of [23].Q.E.D.Saturated mappings.Let us assume that we have a topologically conjugate pair f andˆf, f, ˆf ∈F. Let fi and ˆfi be the corresponding renormalization sequences.

As a consequenceof f and ˆf being conjugate, fi and ˆfi are conjugate for each i. Also, both renormalizationsequences are of the same length.Definition 1.5 Let a renormalization sequence fi be given, and let i < ω.

Then, we definethe saturated map φi of fi as a generalized induced map (Definition 1.2) on the fundamentalinducing domain of fi.The domain of φi is the the backward orbit of the fundamentalinducing domain J of fi+1 under fi. Restricted to a connected set of points whose first entrytime into J is j, the mapping is f j.Definition 1.6 If i < ω, a saturated branchwise equivalence υi is any branchwise equiva-lence between the saturated maps.

If i = ω, the saturated branchwise is also defined equiva-lence and is just the topological conjugacy on the fundamental inducing domain of fi.In this situation, we have a following fact:Fact 1.3 Let f and ˆf be a topologically conjugate pair of renormalizable mappings with theirrenormalization sequences. Assume the existence of a K > 0 and a sequence of saturatedbranchwise equivalences υi, i ≤ω which satisfy these estimates.• Every υi is K-quasisymmetric.• For i < ω every domain of υi is adjacent to two other domains, and for any pair ofadjacent domains the ratio of their lengths is bounded by K.7

• For every i < ω all branches of the corresponding saturated maps are at least 1/K-extendible.Then, the topological conjugacy between f and ˆf is quasisymmetric with a norm boundedas a function of K only.Proof:This is a direct consequence of Theorem 2 of [18].Q.E.D.Further reduction of the problem.We will prove the following theorem:Theorem 1Suppose that f and ˆf are both in Fη for some η > 0 and are topologically conjugate. Then,there is a bound K depending only on η so that there is a K-quasisymmetric saturatedbranchwise equivalence υ0.

In addition, if ω > 0, the all branches of the saturated maps φ0and ˆφ0 are 1/K-extendible.Theorem 1 implies the Reduced Theorem.We check the hypotheses of Fact 1.3.To check that υi is uniformly quasiconformal, we apply Theorem 1 to fi (after an affinechange of coordinates the resulting map is in F). By Fact 1.1, all these mappings belong tosome F˜η where ˜η only depends on η.

So, Theorem 1 implies that all saturated mappings areuniformly quasisymmetric. In the same way we derive the uniform extendibility of saturatedmaps φi and ˆφi.

It is a well-known fact (see [12]) that preimages of the fundamental inducingdomain of fi+1, or of any neighborhood of 0, by fi are dense.We still need to check the condition regarding adjacent domains of φi and ˆφi. We onlydo the check for φi, since it is the same in the phase space of the other mapping.

Inside thedomain of fi+1 every domain of φi is adjacent to two others with comparable lengths. Forthis, see the proof of Proposition 1 in [18] where the preimages of the fundamental inducingdomain are explicitly constructed.

The computations done there are also applicable in ourcase since the “distortion norm” used there is bounded in terms of ˜η by the Real K¨obeLemma.Denote by W the maximal restrictive interval of fi (so W is the domain of fi+1). Outsideof W, consider a connected component of point with the same first entry time j0 into theenlargement of W with scale 1 + ˜η.By this definition and the Real K¨obe Lemma, thedistortion of f j0 on f −j0i(W) is bounded in terms of ˜η.

Now every pair of adjacent domains ofφi can be obtained as the image under some f −j0 in this form of a pair of adjacent domainsfrom the interior of the restrictive interval of fi. It follows that the condition of Fact 1.3regarding the ratio of lengths of adjacent domains is satisfied for any pair.

Now, the ReducedTheorem follows from Theorem 1.Theorem 1 for non-renormalizable mappings.Theorem 1 also has content fornon-renormalizable mappings. It follows from [28] if f and ˆf are in the quadratic family, orrenormalizations of quadratic polynomials.

Our Theorem 1 is marginally more general.8

1.5Box mappingsReal box mappings.Definition 1.7 A generalized induced map φ on an interval J symmetric with respect to 0is called a box mapping, or a real box mapping, if its branches satisfy these conditions. Forevery branch consider the smallest interval W symmetric with respect to 0 which contains therange of this branch.

We will say that W is a box of φ, and we will also say that the branchranges through W. For every branch of φ, it is assumed that it ranges through an intervalwhich is contained in J, and that the endpoints of this interval are not in the domain U ofφ. In addition, it is assumed that all branches are monotone except maybe the one whosedomain contains 0 and that there is a monotone branch mapping onto J.

Also, the centraldomain of φ is always considered a box of φ.Every box mapping has a box structure which is simply the collection of all its boxesordered by inclusion.Complex box mappings.Definition 1.8 Let φ be a real box mapping. We will say that Φ is a complex box mappingand a complex extension of φ provided that the following holds:• Φ is defined on an open set V symmetric with respect to the real axis.

We assume thatconnected components of V are topological disks, call them domains of Φ, and referthe restriction of Φ to any of its domains as a branch of Φ, or simply a complex branch.• If a domain of Φ has a non-empty intersection W with the real line, then W must bea domain of φ in the sense of Definition 1.7. Moreover, the complex branch definedon this domain is the analytic continuation of the branch of φ defined on W, and thisanalytic continuation has no critical points in the closure of the domain except on thereal line.• Every domain of φ which belongs to a branch that does not map onto the entire J isthe intersection of the real line with a domain of Φ.

If this is true for those domainsbelonging to long monotone branches as well, we talk of a complex box mapping withdiamonds.• If two domains of Φ are analytic continuations of branches of φ that range through thesame box of φ, then the corresponding two branches of Φ have the same image.• If a domain of Φ is disjoint with the real line, then the branch defined there is univalentand shares its range with some branch of Φ whose domain intersects the real line.• The boundary of the range of any branch of Φ is disjoint with V .We see that a complex box mapping also has a box structure defined as the set of theranges of all its branches. These complex boxes are in a one-to-one correspondence with theboxes of the real mapping φ.

Also, we will freely talk of monotone or folding branches forcomplex mappings, rather than univalent or degree 2.9

Special types of box mappings.The definitions below have the same wording for realand complex box mappings. So we just talk of a “box mapping” with specifying whether itis real or complex.Definition 1.9 A box mapping φ is said to be of type I if it satisfies these conditions.• The box structure contains three boxes, which are denoted B0 ⊃B′ ⊃B.• B equals the domain of the central branch (i.e.

the only domain of φ which contains0. )• Every monotone branch maps onto B or B0.

Depending on which possibility occurs,we talk of long and short monotone branches.• Every short monotone branch can be continued analytically to a diffeomorphism ontoB′ and the domain of such a continuation is either compactly contained in B′, or com-pactly contained in the complement of B′. Restricted to the real line, this continuationhas negative Schwarzian derivative.• The central branch is folding and ranges through B′.

It also has an analytic contin-uation of degree 2 which maps onto a larger set B′′ ⊂B0 which compactly containsB′. The domain of this continuation is compactly contained in B′.

The Schwarzianderivative of this continuation restricted to the real line is negative.• The closure of the union of domains of all short monotone branches is disjoint with theboundary of B′.• If the closures of two domains domains intersect, at least one of these domains is longmonotone.A real box mapping of type I is shown on Figure 1. A complex box mapping of type I isshown on Figure 2.Definition 1.10 A box mapping φ is said to be of type II if the following is satisfied.• The box structure contains three boxes, B0 ⊂B′ ⊃B.

B denotes the domain of thecentral branch.• All monotone branches of φ map either onto B0 or B′, and are accordingly classifiedas long or short.• The central branch is folding, and it has an analytic continuation whose domain iscompactly contained in B′, the range is some B′′ ⊂B0 which compactly contains B′,the degree of this continuation is 2, and the Schwarzian derivative of its restriction tothe real line is negative.• The closure of the union of all domains of short monotone branches is disjoint with theboundary of B′.10

• If the closures of two domains domains intersect, at least one of these domains is longmonotone.Definition 1.11 A box mapping will be called full if it has only two boxes, B0 ⊃B. Also,the central branch must be folding and range through B0.Hole structures.Given a complex box mapping Φ, the domains which intersect the realline and such that the range of Φ restricted to any of them is less than the largest box will becalled holes.

The central domain is also considered a hole, and described as the central hole.Other domains of Φ which intersects the real line will be called diamonds. Boxes, holes anddiamonds can then be studied purely geometrically, without any reference to the dynamics.So we consider hole structures which are collections of boxes and holes, and hole structureswith diamonds which also include diamonds.

Given a hole structure, the number of its boxesminus one will be called the rank of the structure.Some geometry of hole structures.Although the facts which we will state nowhave an easy generalization for hole structures of any rank, we formally restrict ourselves tostructures of rank 0. We want to specify what bounded geometry is for a hole structure.

Apositive number K is said to provide the bound for a hole structure if the series of estimateslisted below are satisfied. Here we adopt the notation I to mean the box of the structure.1.

All holes are strictly contained in I, moreover, between any hole and I there is anannulus of modulus at least K−1.2. All holes and I are at least K-quasidisks.

The same estimate holds for half-holes andhalf-I, that is, regions in which a half of the quasidisks was cut offalong the real line.3. Consider a hole and its intersection (a, b) with the real line.

Then, there are pointsa′ ≤a and b′ ≥b with the property that the hole is enclosed in the diamond of angleπ/2 −K−1 based on a′, b′. Furthermore, for different holes the corresponding intervals(a′, b′) are disjoint.The “mouth lemma”.Definition 1.12 A diamond neighborhood D(θ) of an interval J given by an angle 0 < θ <π is the union of two regions symmetric with respect to the real line and defined as follows.Each region is bounded by a circular arc which intersects the real line in the endpoints of J.We adopt the convention that θ close to 0 given a very thin diamond, while θ close to π givessomething called a “butterfly” in [27].We know consider an enriched hole structure of rank 0.Definition 1.13 Given a rank 0 hole structure, in the part of J which is not covered by theholes we arbitrarily choose a set of disjoint open intervals.

For each interval, we define adiamond neighborhood. Each diamond has a neighborhood of modulus K−1hwhich is contained11

in the box. Next, each diamond neighborhood contains two symmetrical arcs.

It is assumedthat each of them joins the endpoints in the interval and except for them is disjoint with theline, and that each arc is Kh-quasiarc. This object will be referred to as a hole structure withdiamonds.A bound for a hole structure with diamonds is the greater of the norm for the hole structurewithout diamonds and Kh.For a hole structure with diamonds, consider the “teeth”.

i.e. the curve which consists ofthe upper halves of the boundaries of the holes and the diamonds, and of points of the realline which are outside of all holes and diamonds.

Close this curve with the “lip” which meansthe upper half of the boundary of I. The whole curve can justly be called the “mouth”.Lemma 1.1 For a bounded hole structure with diamonds, the mouth is a quasidisk with thenorm uniform with respect to the bound of the map.Proof:The proof is based of the three point property of Ahlfors (see [1].) A Jordan curve is said tohave this property if for any pair of its points their Euclidean distance is comparable withthe smaller of the diameters of the arcs joining them.

Moreover, a uniform bound in thethree point property implies an estimate on the distortion of the quasicircle. Our conditionsof boundedness were set precisely to make it easy to verify the three point property.

One hasto consider various choices of the two points. If they are both on the same tooth, or both onthe lip, the property follows immediately from the fact that teeth are uniform quasicircles.An interesting situation is when one point is on one tooth and the other one on anothertooth.

We notice that it is enough to check the diameter of a simpler arc which goes alongeither tooth to the real line, than takes a shortcut along the line, and climbs the other tooth(left to the reader.) Suppose first that both teeth are boundaries of holes.

Consider the onewhich is on the left. We first consider the arc which goes to the b′.

By conditions 2 and3 of the boundedness of hole structures, the diameter of this arc is not only comparable tothe distance from the point to b′, but even to the distance from the projection of the pointto the line. The estimate follows.

The case when one of the teeth is a diamond is left tothe reader. Another interesting case is when one point is on the lip and another one on atooth.

If the tooth is a diamond, the estimate follows right away from the choice of angles.If the tooth is a hole, we have to use condition 1. We use the bounded modulus to constructa uniformly bounded quasiconformal map straightens the lip and the tooth to round circleswithout changing the modulus too much.

The estimate also follows.Q.E.D.Extension lemma.We are ready to state our main result which in the future willbe instrumental in extending real branchwise equivalences to the complex domain. Giventwo hole structures with diamonds, a homeomorphism h is said to establish the equivalencebetween them if it is order preserving, and for any hole, box, or diamond of one structure,it transforms its intersection with the real axis onto the intersection between a hole, box ordiamond respectively of the other structure with the real axis.12

Lemma 1.2 For a uniform constant K, let two K-bounded hole structures with diamonds bewith an equivalence establishing K-quasisymmetric homeomorphism h. We want to extendh to the complex plane with prescribed behavior on the boundary of each tooth and on thelip. This prescribed behavior is restricted by the following condition.

For each tooth or thelip consider the closed curve whose base is the interval on the real line, and the rest is theboundary of the tooth or the lip, respectively. The map is already predefined on this curve.We demand that it maps onto the corresponding object of the other hole structure, and thatit extends to a K-quasiconformal homeomorphism.

Then, h has a global quasiconformalextension with the prescribed behavior and whose quasiconformal norm is uniformly boundedin terms of K.Proof:We first fill the teeth with a uniformly quasiconformal map which is something we assumedwas possible. Next, we extend to the lower half-plane by [3].

Then, we want an extensionto the region above the lip and above the real line. This is certainly possible, since theboundary of this region is a uniform quasicircle (because the lip was such, use the threepoint property to check the rest).

But the map can be defined below this curve by filling thelip and using [3] in the lower half-plane. Then one uses reflection (see [1].) Now, the mapis already defined outside of the mouth, one uses Lemma 1.1 and quasiconformal reflectionagain to finish the proof.Q.E.D.1.6Introduction to inducingStandard extendibility of real box mappings.For monotone branches we have thenotion of extendibility which is compatible with our statement of the Real K¨obe Lemma(Fact 1.1.) That is, a monotone branch with the range equal to (b, c) will be deemed ǫ-extendible if it has an extension as a diffeomorphism with negative Schwarzian derivativeonto a larger interval (a, d) and so that|a −b| · |d −c||c −a| · |d −b| ≥ǫ .This amounts to saying that the length of (b, c) in the Poincar´e metric of (a, d) is no morethan −log ǫ.

1 The interval (a, d) will be called the margin of extendibility while the domainof the extension will be referred to as the collar of extendibility.We can also talk of extendibility for the central folding branch. To this end, we representthe folding branch as h(x2).

Suppose that the central branch ranges through a box (b, c).Then the folding branch is considered ǫ-extendible provided that h has an extension as adiffeomorphism with negative Schwarzian derivative onto a larger interval (a, d) and thePoincar´e length of (b, c) inside (a, d) is at most −log ǫ. Again, (a, d) is described as themargin of extendibility and its preimage by the extension of the folding branch is the collarof extendibility.1See [23] for a discussion of the Poincar´e metric on the interval.13

Definition 1.14 Consider a box mapping φ on J. Standard ǫ-extendibility for φ meansthat the following properties hold.• There is an interval I ⊃J so that the Poincar´e length of J in I is no more than−log ǫ.

Every branch of φ which ranges through J is extendible with the margin ofextendibility equal to I. Furthermore, for each such branch the collar of extendibilityalso has Poincar´e length not exceeding ǫ in I; moreover if the domain of the branch iscompactly contained in J, so is the the collar of extendibility.• For every other box B in the box structure of φ, there is an interval IB ⊃B whichserves as the margin of extendibility for all branches that range through B. IB must becontained in any box larger than B.

If the domain of a branch is contained in B, thecollar of extendibility of this branch must be contained in IB, moreover, if this domainis compactly contained in B, the collar is contained in B.We will call a choice of I and IB the extendibility structure of φ.Refinement at the boundary.Consider a generalized induced map φ on J. If we look atan endpoint of J, clearly one of two possibilities occurs.

Either the domains of φ accumulateat the endpoint, or there is one branch whose domain has the endpoint on its boundary.In the first case, we will say that φ is infinitely refined at its (left, right) boundary. In thesecond case the branch adjacent to the the endpoint will be called the (left, right) externalbranch.Assume now that an external branch ranges through J, and that its common endpointwith J is repelling fixed point of the branch.

Call the external branch ζ. We will describethe construction of refinement at a boundary.

Namely, we can define φ1 as φ outside of thedomain of ζ, and as φ◦ζ on the domain of ζ. Inductively, φi+1 can be defined as φ outside ofthe domain of ζ, and φi ◦ζ on the domain of ζ. The endpoint is repelling.

Thus, the externalbranches of φi adjacent to the boundary point will shrink at an exponential rate. We caneither stop at some moment and get a version of φ finitely refined at the boundary or proceedto an L∞limit of this construction to obtain a mapping infinitely refined at this boundary.There is an analogous process for the other external branch, ζ′ (it is exists).

Namely, define(φi)′ as φ outside the domain of ζ′ and φi−1 ◦ζ′ on the domain of ζ′.We call the mapping obtained in this process a version of φ refined at the boundary(finitely or infinitely.) Observe that the refinement at the boundary does not change eitherthe box structure of the map or its extendibility structure.Operations of inducing.Loosely speaking, inducing means that some branches of a boxmapping are being replaced by compositions with other branches.

Below we give precisedefinitions of five procedures of inducing.Simultaneous monotone pull-back.Suppose that a box mapping φ on J is givenwith a set of branches of φ which range through J. We denote φ′ = φb where φb is a versionof φ refined at the boundary.

Then on each branch ζ from this set replace φ with φtbζ whereφtb is φb with the central branch branch replaced by the identity. Observe that this operation14

does not change the box or extendibility structures. This is a tautology except for the onewhich is just the restriction of ζ to the preimage of the central domain.

This is extendiblewith margin I, so the standard extendibility can be satisfied with the same ǫ.Filling-in.To perform this operation, we need a box mapping φ and φ′ with the samebox and extendibility structures as φ and with a choice of boxes B1 smaller than B0 := Jbut bigger than the central domain B. We denote φ0 := φ′ and proceed inductively byconsidering all monotone branches of φ which map onto B1.

On the domain of each suchbranch ζ, we replace ζ with φtiζ. Here, φti denotes φi whose central branch was replaced bythe identity.

The resulting map is φi+1. In this way we proceed to the L∞limit, and thislimit φ∞is the outcome of the filling-in.

Note that B1 drops from the box structure of φ∞,but otherwise the new map has the the same boxes with the same margins of extendibilityas φ. Thus, if φ satisfied standard ǫ-extendibility, so does φ∞.Simple critical pull-back.For this, we need a box mapping φ whose central branchis folding, and another box mapping φ′ about which we assume that it has the same boxstructure as φ and is not refined at either boundary.

Let φt denote φ′ whose central branchwas replaced by the identity, and ψ be the central branch of φ. Then the outcome is equalto φ modified on the central domain by substituting ψ with φt ◦ψ.Almost parabolic critical pull-back.The arguments of this operation are the sameas for the simple critical pull-back.

In addition, we assume that the critical value of ψ is inthe central domain of ψ (and ψ′), but 0 is not in the range of ψ from the real line. Also,it is assumed that ψl(0) /∈B so some l > 0, and let l denote the smallest integer with thisproperty.

Then, for each point x ∈B we define the exit time e(x) as the smallest j so thatψj(x) /∈B. Clearly, e(x) ≤l for every x ∈B.

Then the outcome of this procedure is ψmodified on the subset of points with exit times less than l by replacing ψ withx →ψ′ ◦ψe(x)(x) .On the set of point with exit time equal to l, which form a neighborhood of 0, the map isunchanged.Critical pull-back with filling-in.This operation takes a box mapping φ which mustbe of type I or full, and another mapping φ′ with the same box structure and not refined ateither boundary. Let ψ denote the central branch of φ, and let χ be the generic notation fora short monotone branch of φ.

We define inductively two sequences, φi and φri. Let φ0 := φand φr0 be φ′ with the central branch replaced by the identity.

Then φi+1 is obtained from φby replacing the central branch ψ with φriψ and every short monotone branch χ of φ withφri ◦ψ ◦χ. Then φri+1 is obtained in a similar way, but only those short monotone and foldingbranches of φ are replaced whose domains are contained in the range of ψ.

Others are leftunchanged. At the end, the central branch of the map obtained in this way is replaced bythe identity.

The outcome is the L∞limit of the sequence φi. This is the most complicatedprocedure and is described in [18], and called filling-in, in the case when φ is full.

We note15

here that φ∞is either of type I, full or Markov depending on the position of the critical valueof ψ. If the critical value was in a long monotone domain of φ′, then the outcome is a fullmapping.

If it was in a short monotone domain of φ′ or in the central domain, the result isa type I map. If the critical value was not in the domain of φ′, the result is a Markov map,that is a mapping whose all branches are monotone and onto J.Boundary refinement.The five basic operations described above obviously fall in twoclasses.

The first two involve composing with monotone branches and do not affect standardextendibility. The last three involve composing with folding branches and can affect standardextendibility.

Each of these last three operations can be preceded by a process called boundaryrefinement. To do the boundary refinement, assume that a box mapping φ is given whosecentral branch ψ is folding.

Then look at the set of “bad” branches of φ. For the stepsof simple critical pull-back and critical pull-back with filling-in, a branch of φ is considered“bad” if it ranges through J, its domain is in the range of ψ and its collar of standardextendibility contains the critical value.For almost parabolic critical pull-back the lastcondition is weakened so that the branches which contain ψi(0), 0 < i ≤l are also consideredbad.

Then boundary refinement is defined as the simultaneous monotone pull-back on theset of bad branches with φ′ that may vary with the branch. There are two possibilities here.If we do infinite boundary refinement, as φ′ we use the version of φ infinitely refined at bothendpoints.

If we do minimal boundary refinement, we use the version of φ refined at theendpoint of the side of the critical value only, and refined to the minimal depth which willenforce standard extendibility.This gives the mapping φ′ which then enters the corresponding procedure of criticalpull-back.Lemma 1.3 Let φ be a type I or full mapping with standard ǫ-extendibility. Apply criticalpull-back with filling-in preceded by boundary refinement.

Then, the resulting map φ∞alsohas standard ǫ-extendibility.Proof:Boundary refinement assures us that for every i the branches of φi or φri which range throughJ are extendible with the same margin as in the original φ. This ends the proof if φ∞iffull.

Otherwise, let B denote the central domain of φ∞(the same as the central domain ofφi, any i > 0), and B′ the central domain of φ. Observe that all short branches of φ∞areextendible with margin B′.

Thus, one can set IB′ inherited from the extendibility structureof φ, and IB := B′. One easily checks that the conditions of standard extendibility hold.Q.E.D.16

2Initial inducing2.1Real branchwise equivalencesStatement of the result.Our definitions follow [18].Definition 2.1 Let υ1 be a quasisymmetric and order-preserving homeomorphism of the lineonto itself. Let υ2 be a quasisymmetric order-preserving homeomorphism from an interval Jonto another interval, say J′.

We will say that υ2 replaces υ1 on an interval (a, b) with distor-tion L if the following mapping υ is an L-quasisymmetric order preserving homeomorphismof the line onto itself:• outside of (a, b), the mapping υ is the same as υ1,• inside (a, b), υ has the formυ = A′ ◦υ2 ◦A−1where A and A′ are affine and map J onto [a, b] and J′ onto υ1([a, b]) respectively.Definition 2.2 We will say that a branchwise equivalence υ between box mappings on Jsatisfies the standard replacement condition with distortion K provided that• υ restricted to any domain of the box mapping replaces υ on J,• υ restricted to any box of this mapping replaces υ on J.Definition 2.3 A box map φ is said to be α-fine if for every domain D and box B so thatD ⊂B but D ̸⊂B|D|dist(D, ∂B) ≤α .Definition 2.4 A branch is called external in the box B provided that the domain D ofthe branch is contained in B, but the closure of D is not contained in D. We will say thatthe map is not refined at the boundary of B provided that an external branch exists at eachendpoint of B.Proposition 1 Suppose that box mappings φ and φ′ are given which will enter one of the fiveinducing operations defined in section 1.6. Assume further that ˆφ is topologically conjugateto φ, and ˆφ′ is topologically conjugate to φ′, with the same topological conjugacy.

Let υ0 andυ′0 be the branchwise equivalences between φ and ˆφ as well as φ′ and ˆφ′ respectively. Supposethat all these branchwise equivalences coincide outside of J.

Assume in addition that• all branches of φ, φ′, ˆφ, ˆφ′ are ǫ-extendible,• all four box mappings are α-fine,17

• υ0 and υ′0 satisfy the standard replacement condition with distortion K and are Q-quasisymmetric,• φ′ and ˆφ′ are not refined at the boundary of any box smaller than B0,• if D is the domain of an external branch of in a box B (for φ′ or ˆφ′), then |D|/|B| ≥ǫ1.Then for all ǫ, K, Q, α and ǫ1 there are bounds L1 and L2 so that the following holds.Perform one of the five inducing operations on φ and φ′ as well as ˆφ and ˆφ′.Call theoutcomes φ∞and ˆφ∞, or φ∞,b and ˆφ∞,b for their versions refined at the boundary. Then thereare branchwise equivalences υ∞and υ∞,b between φ∞and ˆφ∞, or φ∞,b and ˆφ∞,b respectivelywhich satisfy• υ∞and υ∞,b are the same as υ outside of J,• υ∞and υ∞,b satisfy the standard replacement condition with distortion L1,• υ∞and υ∞,b are both L2-quasisymmetric.Some technical material.The proof of Proposition 1 splits naturally in five cases de-pending on the kind of inducing operation involved.Among these the almost paraboliccritical pull-back stands out as the hardest.

Proposition 1 in all remaining cases followsrather straightforwardly from the work of [18]. We begin by recalling the technical toolsof [18].Definition 2.5 Given an interval I on the real line, its diamond neighborhood is a setsymmetrical with respect to the real axis constructed as follows.

In the upper half-plane, thediamond neighborhood is the intersection of a round disk with the upper half-plane, while onthe intersection of the same disk with the real line is I. For a diamond neighborhood, itsheight refers to twice the Hausdorffdistance between the neighborhood and I divided by thelength of I.This is slightly different from the definition used in [18], but this is not an essentialdifference in any argument.

The nice property of diamond neighborhoods is the way theyare pull-back by polynomial diffeomorphisms.Fact 2.1 Let h be a polynomial which is a diffeomorphism from an interval I onto J. Assumethat h preserves the real line and that all its critical values are on the real line. Suppose alsothat h is still a diffeomorphism from a larger interval I′ ⊃I onto a larger interval J′ ⊃Jso that the Poincar´e length of J inside J′ does not exceed −log ǫ.

Then, there are constantsK1 > 0 and K2 depending on ǫ only so that the distortion of h (measured as |h′′/(h′)2|) onthe diamond neighborhood of I with height K1 is bounded by a K2.Proof:Observe that the inverse branch of h which maps J′ onto I′ is well defined in the entire slitplane C \ J′. Since h is a local diffeomorphism on I, the inverse branch can be defined ona diamond neighborhood of J′ with sufficiently small height.

As we gradually increase the18

height of this neighborhood we see that the inverse branch can be defined until the boundaryof the neighborhood hits a critical value of h. That will never occur, though, since all criticalvalues are on the real line by assumption. Now, the diamond neighborhood of J with height1 is contained in C \ J′ with modulus bounded away from 0 in terms of ǫ.

So, by K¨obe’sdistortion lemma, the preimage of this diamond neighborhood by the inverse branch of hcontains a diamond neighborhood of I of definite height, and the distortion of h there isbounded.Q.E.D.Pull-back of branchwise equivalences.Definition 2.6 A pull-back ensemble is the conglomerate of the following objects:• two equivalent induced mappings, one in the phase space of f, the other in the phasespace of ˆf, with a branchwise equivalence Y and a pair of distinguished branches ∆and ˆ∆whose domains, D and ˆD respectively, correspond by Y,• a branchwise equivalence Υ which must map the critical value of ∆to the critical valueof ˆ∆in case if ∆is folding.The following objects and notations will always be associated with a pull-back ensemble.If ∆is monotone, let (−Q, Q′), 65Q > Q′ ≥Q, be its range. If ∆is folding, make Q′ = Qand define (−Q, Q′) to be the smallest interval symmetrical with respect to 0 which containsthe range of ∆.

Analogously, we define ( ˆQ, ˆQ′) to be the range of ˆ∆or the maximal intervalsymmetric with respect to 0 containing the range of ˆ∆, and we also assume that 65 ˆQ > ˆQ′ ≥ˆQ.In addition, the following assumptions are a part of the definition:• Υ((−Q, Q′)) = (−ˆQ, ˆQ′),• Υ(z) = Υ(z) and Υ(−z) = −Υ(z) for every z ∈C,• outside of the disc B(0, 43Q) the map Υ has the formz →ˆQ′ −ˆQQ′ −Qz ,• the mapping Υ restricted to the setR \ (−Q, Q′)has a global λ-quasiconformal extension,• the branches ∆and ˆ∆are ǫ-extendible,• if ∆is folding, then its range is smaller than (−Q, Q′), but the length of the range isat least ǫ1Q; likewise, if ˆ∆is folding, then its range does not fill the interval (−ˆQ, ˆQ′),but the length of the range is at least ǫ1 ˆQ.The numbers ǫ, ǫ1 and λ will be called parameters of the pull-back ensemble.19

Vertical squeezing.Definition 2.7 Suppose that two parameters s1 ≥2s2 > 0 are given. Consider a differen-tiable monotonic function v : R →R with the following properties:• v(−x) = −x for every x,• if 0 < x < s2, then v(x) = x,• if s1 < x, then v(x) = x −s1 + 2s2.Each v defines a homeomorphism of the plane V defined byV(x + iy) = x + iv(y) .We want to think of V as being quasiconformal end depending on s1 and s2 only.

To thisend, for a given s1 and s2 pick some v which minimizes the maximal conformal distortionof V. We will call this V the vertical squeezing map for parameters s1 and s2.The Sewing Lemma.Fact 2.2 Consider a pull-back ensemble with Q = Q′. Let Y be a branchwise equivalence withD as a domain, and suppose that restricted to D it replaces Υ on (−Q, Q) with distortionM.

Suppose that for some R > 0 the map Υ transforms the diamond neighborhood withheight R of (−Q, Q) exactly on the diamond neighborhood with height R of (−ˆQ, ˆQ). Chooser > 0 and C ≤1 arbitrary and assume that the diamond neighborhood with height r of Dis mapped by Y onto the diamond neighborhood with height Cr of ˆD Assuming that R ≤R0where R0 only depends on the parameter ǫ of the pull-back ensemble, for every such choice ofC, M, r, R, and a set of parameters of the pull-back ensemble, numbers K and L, parameterss1, s2, ˆs1, ˆs2, a mapping ˜Υ as well as a branchwise equivalence ˜Y exist so that:• if V is the vertical squeezing with parameters s1, s2, and ˆV is the vertical squeezing mapwith parameters ˆs1 and ˆs2, then ˜Y has the form˜Y = ˆV ◦˜Υ ◦V−1on the image of the domain of ˜Υ by V,• the domain and range of ˜Υ are contained in diamond neighborhoods with height 1/10of D and ˆD respectively,• on the diamond neighborhood with height K of D˜Υ = ∆−1 ◦Υ ◦∆which means the lift to branched covers, order-preserving on the real line in the case of∆folding,20

• outside of this diamond neighborhood the conformal distortion of ˜Υ is bounded by thesum of L and the conformal distortion of Υ outside of the image of the diamondneighborhood of D with height K by ∆,• ˜Y coincides with Y outside of the diamond neighborhood with height r of D,• on the set-theoretical difference between the diamond neighborhoods with heights r andr/2, the map ˜Y is quasiconformal and its conformal distortion is bounded as the sumof L and the conformal distortion of Y on the diamond neighborhood of D with heightr,• outside of the diamond neighborhood with height r/2 of D, the mapping ˜Y is indepen-dent of Υ,• on the set-theoretical difference between the diamond neighborhood of D with height r/2and U the map ˜Y has conformal distortion bounded almost everywhere by L.Proof:This fact is a consequence of Lemmas 4.3 and 4.4 of [18]. The context of [18] is slightlydifferent from our situation, since that paper works in the category of S-unimodal mappingsand uses their “tangent extensions” instead of analytic continuation.

However, proofs givenin [18] work in our situation with only semantic modifications, so we do not repeat themhere.Q.E.D.As a consequence of the Sewing Lemma, we getFact 2.3 Suppose that a pull-back ensemble is given so that Υ transforms the diamondneighborhood with height R of (−Q, Q) exactly to the homothetic diamond neighborhood of(−ˆQ, ˆQ). Assume also that R ≤R0 as required by the hypothesis of the Sewing Lemma, andthat Y restricted to D replaces Υ on (−Q, Q) with distortion M. Suppose finally that bothΥ and Y are Q-quasiconformal.

Then there is a map ˜Y which differs from Y only on thediamond neighborhood of D with height 1/2, is equal toˆ∆−1 ◦Υ ◦∆on D, and is L-quasiconformal. The number L only depends on M, Q and the parametersof the pull-back ensemble.Proof:This follows directly from Fact 2.2 when one chooses r = 1/2 and observes that C is boundedas a function of Q.Q.E.D.21

Beginning the proof of Proposition 1.In order to be able to use Fact 2.2 we need alemma that will allow us to build pull-back ensembles.Lemma 2.1 Suppose that a branchwise equivalence υ between topologically conjugate boxmappings φ and ˆφ is given on (−Q, Q).Assume that this branchwise equivalence is Q-quasisymmetric and satisfies the standard replacement condition with distortion K. Supposealso that a pair of branches ∆and ˆ∆are given so that ∆and ˆ∆range through boxes thatcorrespond by υ. Moreover, assume that if ∆one is folding, the other one is, too, and thecritical values correspond by the topological conjugacy.

Suppose also that φ and ˆφ are bothα-fine and all of their branches are ǫ-extendible. If ∆and ˆ∆are also ǫ extendible, then therethere is a mapping υ′ which coincides with υ on (−65Q, 65Q) except on the domain whichcontains the critical value of ∆, and an L-quasiconformal extension of υ′, called Υ, whichtogether with ∆and ˆ∆gives a pull-back ensemble with parameters ǫ, ǫ1 and λ. NumbersL and λ depend on Q and K only.

The mapping Υ transforms the diamond neighborhoodwith height R0 of (−Q, Q) exactly onto the diamond neighborhood of (−ˆQ, ˆQ) with the sameheight, where R0 is determined in terms of ǫ by Fact 2.2.Proof:We have two tasks to perform: one is to build υ′ so that it maps the critical value of ∆tothe critical value of ˆ∆, and second is to construct the proper quasiconformal extension. Letus address the second problem first.

We use this fact.Fact 2.4 Let υ be a Q-quasisymmetric mapping of the line into itself, and let (−ˆQ, ˆQ) =υ((−Q, Q)). For every R < 1 there an L-quasiconformal homeomorphism Υ of the planewith these properties:• Υ maps the diamond neighborhood of (−Q, Q) with height R exactly onto the diamondneighborhood of (−ˆQ, ˆQ) with the same height,• Υ restricted to (−65Q, 65Q) equals υ,• Υ maps B(0, 43Q) exactly onto B(0, 43 ˆQ) and is affine outside of this ball.The bound L depends on R (continuously) and Q only.Fact 2.4 follows from the construction done in [18] in the proof of Lemma 4.6.

FromFact 2.4 we see that once υ′ is constructed with the desired properties, Lemma 2.1 follows.To get υ′, we first construct take any point c and construct υ′′ which maps c to H(c),where H is the topological conjugacy, and υ′′ = υ outside of the box that contains c. Toconstruct υ′′, we consider two cases. If the c is not in an external domain, one can composeυ′′ with a diffeomorphism of bounded distortion which moves υ(c) to H(c) and is the identityoutside of the box.

If c is an external domain, map it by the external branch into the phasespace of a version of φ′ refined at the boundary. By the previous argument, this imagerequires only a push by a diffeomorphism of bounded distortion.

This way we get somebranchwise equivalence υ1. Using Fact 2.4, we build the pull-back ensemble which consistsof the external branch, its counterpart in the phase space of ˆφ′, and the appropriate extensionΥ1 of υ1.

To this pull-back ensemble we can apply Fact 2.3 and get υ′′ on the real line. By22

the way, in this case υ′′ can immediately be taken as υ′. Generally, in order to obtain υ′from υ′′ we apply a similar procedure.

Take the branch which contains the critical value of∆and use the image of the critical value by this branch as the c to build υ′′. Then constructthe pull-back ensemble by Fact 2.4 and pull υ′′ back to get υ′.

Again, υ′ is quasisymmetricfrom Fact 2.3Q.E.D.We will next prove that Lemma 2.1 and the Sewing Lemma allow us to construct υ∞andυ∞,b which are quasisymmetric as needed. The construction of υ∞,b is quite the same as forυ∞, only we start with φb.

So we only focus on the construction of υ∞,b.Bounded cases of inducing.Suppose that we are in the situation of Proposition 1. In thecase of a simple critical pull-back and simultaneous monotone pull-back, the quasisymmetricestimate for υ∞follows directly.

By Lemma 2.1 for each branch being refined we constructa pull-back ensemble, and get ˜Y which is quasiconformal with an appropriate bound byFact 2.3. Note that in the case of the simultaneous pull-back the operations on variousbranches do not interfere, since each modifies Y only in the diamond neighborhood withheight 1/2.The case of filling-in.We begin by constructing the extension Υ of υ′0 using Lemma 2.1.For this, choose R equal to R0 which is given by the Sewing Lemma depending on ǫ, and(−Q, Q) equal to the box B1.

Denote Υ0 = Υ. Also, take as Y any quasiconformal exten-sion of υ0 with a norm bounded as a function of Q.

We will proceed inductively and obtainΥi+1 by applying the Sewing Lemma always with the same Y, ∆ranging over the set of allbranches which map onto B1, and Υi.Let δ be the generic notation for the domains of branches of φ mapping onto B1, and letδ−m denote similar domains of φtm. Let us choose r so small that the diamond neighborhoodsof domains δ of φ are contained in the diamond neighborhood with height R of (−Q, Q).In our inductive construction, this will assure that Υi for any i is the same as Υ on theboundary of the diamond neighborhood of (−Q, Q) with height R, so that this diamond isalways mapped on the homothetic diamond.

In particular, parameters ǫ, λ and C remainfixed since only the Υ component changes in pull-back ensembles. Note that C is determinedby the conformal distortion of Y, i.e.

by the parameter Q. Also, since Υi−1 coincides with Υon the real line outside of (−Q, Q) the condition that Υ restricted to a domain replaces Υi−1on (−Q, Q) is satisfied with the same distortion M. Thus, for all i the pull-back ensembleused to construct Υi satisfies the hypotheses of the Sewing Lemma with the same parameters.So we will regard estimates claimed by these Lemmas as constants.Next, look at the neighborhood of D where Υi has the formΥi = ˆV ◦ˆ∆−1 ◦Υi−1 ◦∆◦V−1 .By Fact 2.1, this contains a diamond neighborhood with height K which is mapped by ∆withbounded distortion.

In particular, by choosing r possibly even smaller, but still controlledby K, we can make sure that the diamond neighborhoods with height r of all domains δ−023

are inside the image of this region (called the inner pull-back region) by ∆. Now, addressthe issue of where Υi−1 and Υi differ.

If i = 1, they differ only on the union of diamondneighborhoods with height r of domains δ−0. For i > 1, Υi and Υi−1 differ on the preimageof the set where Υi−2 and Υi−1 differ by maps∆◦V−1with ∆ranging over the set of all branches mapping onto B1.Next, we see inductively that the set on which Υi and Υi−1 differ is contained in the unionof diamond neighborhoods with fixed height of domains δ−i+1 of φi−1.

This is clearly so fori = 1. In the general case, we need to consider the preimage of the set where Υi−1 differsfrom Υi−2 by ∆◦V−1.

Suppose that Υi−2 and Υi−1 differ only on the union of diamondneighborhoods with height ri−2 of domains δ−i+2. Pick some domain δ−i+2 and observe that∆extends as a diffeomorphism onto B1.

It is easily seen that sizes of domains δ−m decreasewith m at a uniform exponential rate. Therefore that ∆restricted to the preimage of δ isx-extendible with | log x| going up exponentially fast with i.

So, ∆restricted to ∆−1(δ) isalmost affine with distortion going down exponentially fast with i (which follows from K¨obe’sdistortion lemma.) Next, provided that ri−2 < 1, the height of the diamond neighborhoodof δ with height ri−2 with respect to δ0 goes down exponentially with i.

Since the distortiongoes down exponentially fast, the preimage of this diamond neighborhood by ∆with fitinside the diamond neighborhood of δ−i+1 = ∆−1(δ−i+2 with height ri−2(1 + b(i)) where b(i)decreases exponentially fast with i. Thus, by choosing r small enough, we can ensure thatri < 1 for all i.

The same reasoning can be conducted for the phase space of ˆf to provethat the images of these diamonds by Υi are contained in the diamond neighborhoods withheight 1 of the corresponding domains δ−i+1 of ˆφ.For any branch ∆of ϕ, consider the region W defined as the intersection of the innerpull-back region with the set on which V is the identity (i.e. the horizontal strip of width2s2.) This contains a diamond neighborhood with fixed height of the domain of ∆.

So, fori > i0 (i0 depends on how fast the sizes of δ−m decrease with m and can be bounded throughǫ and α), the set on which Υi−2 and Υi−1 differ is contained in the image of W by ∆, andits image by Υi−1 is contained in the image of ˆW byhat∆. By Fact 2.3 applied i0 times, Υi0 is still quasiconformal because each step adds onlya bounded amount of distortion.

For i ≥i0, Υi is obtained on this region by replacing Υi−1withˆ∆−1 ◦Υi−1 ◦∆.This means that for i ≥i0 the conformal distortion of Υi is the same as the conformaldistortion of Υi0. Thus, the limit of Υ∞exists and is quasiconformal with the same boundednorm.Critical pull-back with filling-in.This is very similar to the filling-in case and oncewe have built the initial pull-back ensemble, the proof goes like in Lemma 4.5 of [18].

Weleave the details out.Almost parabolic critical pull-back.This case requires a different set of tools. Let usfirst set up the core problem is abstract terms.24

Definition 2.8 We define an almost parabolic map ϕ be the following properties.• The map ϕ is defined on and interval [0, a) with a < 1.• ϕ(x) = g(x2) where g is an orientation-preserving diffeomorphism onto the image of ϕ,has negative Schwarzian, and is ǫ-extendible, ǫ > 0.• ϕ(a) = 1 and ϕ(x) > x for every x.• On the set (c, a), Sϕ ≥−K.The numbers a, ǫ, K will referred to as parameters of ϕ.Proposition 2 Suppose that two almost parabolic maps ϕ and ϕ′ are given. Suppose that theparameters a are both bounded from below by alpha > 0 and from above by β < 1, likewisethe parameter ǫ is equal for both mappings, and the parameters K are both bounded fromabove by K0.

Consider the finite sequence ai defined by a0 = a and ϕ(ai) = ai−1 for i > 0 orthe sequence ends when such ai cannot be found. Define a′i in the same manner using ϕ′ andassume that the lengths of both sequences are equal to the same l. Define a homeomorphismu from (al, a) onto (a′l, a′) by the requirement that u(ai) = a′i for every i and that u is affineon each segment (ai, ai−1).

Then u is a Q-quasi-isometry with Q only depending on α, β,K0 and ǫ.A comment about Proposition 2.What really matters is that u is quasisymmetric,which follows from its being a quasi-isometry. Proposition 2 is easily accepted by specialistsin the field, perhaps because it is an easy fact when ϕ is known to be complex polynomial-like.However, I was unable to find a fair reference in literature regarding the negativeSchwarzian setting, though I am aware of an unpublished work of J.-C. Yoccoz in whicha similar problem was encountered and solved in the study of critical circle mappings ofunbounded type.

The approach we use here owes to the work [9].Easy bounds.We now assume that ϕ is a mapping that satisfies the hypotheses of Propo-sition 2. Throughout the proof we will refer to bounds that only depend on α, β, ǫ and K0 asconstants.

And so we observe that the derivative of g is bounded from both sides by positiveconstants. This means that the derivative of ϕ is similarly bounded from above.

Next, c isbounded from below by a positive constant, or it would be impossible to maintain ϕ(x) > x.The second derivative of ϕ is also bounded in absolute value by a constant, from the RealK¨obe’s Lemma. Next, there is exactly one point z where ϕ(x) −x attains a minimum.

Thatis since because of the negative Schwarzian there are at most two points where the derivativeof ϕ is one. Also observe that is l is large enough, then the distances from z to ai and a0are bounded from below by constants.

That is because the shortest of intervals (ai, ai−1)is the one containing z, or the adjacent one. If l is large this becomes much smaller thatthe constants bounding from below the lengths of (aj, aj−1) or (al−j, al−j−1) for j < 10.

Asomewhat deeper fact is this.25

Lemma 2.2 There are positive constants l0, K1 and K2 so that if ϕ(x)−x < K1 and l ≥l0,then ϕ′′(x)/ϕ′(x) ≥K2.Proof:The proof almost copies the argument used in the demonstration of Proposition 2 in [9]. Weuse N g to mean g′′/g′.

Take into account the differential equationDN g = Sg + 1/2(N g)2 . (2)which is satisfied by every C3 diffeomorphism g (a direct check, also see [23] page 56.) Thenconsider an abstract class of diffeomorphisms G(w, L) defined as the set of functions definedin a neighborhood of 0 and having the following properties:1. their Schwarzian derivatives are negative and bounded away from 0 by some −β.2.

for any g ∈G, g(0) = w,3. there is no x ∈(−L, L) where g is defined and g(x) ≤xObserve that the functiong0(y) = ϕ(y + x)belongs to G(w, L) withw := ϕ(x) −xand L a constant equal to the minimum of distances from x to c and from x to a.

This is aconstant provided l0 is large enough and K1 is sufficiently small. So, it suffices to show thatfor every L > 0 there is a constant K1 > 0 so that w < K1 implies N g(0) > K2 > 0.We observe that every function from g ∈G(w, L) is uniquely determined by three pa-rameters: a continuous function ψ = Sg and two numbers ν and µ equal to N g(0) andg′(0) respectively.

Indeed, given ψ and ν, N g is uniquely determined by the differentialequation (2), this together with µ determines g′, and finally g is also defined by w. Observethat with µ fixed, g is an increasing function of ψ and ν. Indeed, a look at the equation (2)reveals that if ψ1 ≥ψ2 with the same ν, then the solution N g1(x) ≥N g2(x) for x ≥0 whileN g1 ≤N g2 for x ≤0.

This is immediate if ψ1 > ψ2 since we see that at every point wherethe solutions cross N g1 is bigger on a right neighborhood and less on a left neighborhood.Then we treat ψ1 ≥ψ2 by studying ψ′ = ψ1 + c where c is a positive parameter and usingcontinuous dependence on parameters. As g′ is clearly an increasing function of N g and ν,the monotonicity with respect to ψ and ν follows.

So, if we can show that for some ˜ψ and ˜νand every µ the condition g(x) ≥x is violated on (−L, L), it follows that there exists δ > 0so that for every ψ ≤˜ψ, we must have ν > ˜ν + ǫ if the function is in G(w, L).Pick ˜ψ = −β and ν = 0. The problem becomes quite explicit.

From another well-knowndifferential formula u′′ = Sg · u satisfied by u = 1/sqrtg′ we findg′(x) =µ√cosh βx; .Let wn →0 and pick µn so that the corresponding g satisfies g(x) ≥x, or escapes to +∞,on (−L, L). Observe that µn must be a bounded sequence, since we have g(x) ≤µC(L,β)x+ w26

for x < 0 where C(L, β) is the upper bound of cosh βx on [−L, 0]. Thus if such a sequenceexisted, we could take a limit parameter µ∞which would preserve g(x) ≥x even for w = 0,and this cannot be.Q.E.D.Approximation by a flow.For l large, the condition ϕ(x) −x < K1 of Lemma 2.2 issatisfied on a neighborhood W of z.

On this neighborhood, we can boundB(x −z)2 + δ ≤ϕ(x) −x ≤A(x −z)2 + δwhere δ = ϕ(z)−z. The numbers A and B are constants, and while the upper bound is easyand satisfied in the entire domain of ϕ, the lower one follows from Lemma 2.2 is guaranteedto hold only on W. Note that the number of points ai outside of W is bounded by a constant.So we can change the definition of W a bit so that it is preserved by the map u betweentwo almost parabolic maps.

That is, we can make points am and al−m the endpoints of W,and if m is big enough, the condition ϕ′(x) −x will also hold on u(W). Incidentally, forl large, this means that z′ ∈W ′.

The map u is clearly a quasi-isometry outside of W, sowe have reduced the problem to considering it on W. Let T = l −2m, that is the numberof iterations an orbit needs to travel through W, and let T0 be chosen so that [aT0, aT0+1)contains z. If analogous times are considered for ϕ′, note that T ′ = T, but T0 and T ′0 haveno reason to be the same.Lemma 2.3 Let x1, x2 ∈W with ϕt(x1) = x2.

Then, if z −x1 < K, K constants14Aδ(tan−1(sAδ (x2 −z)) −tan−1(sAδ (x1 −z))) −1 ≤t≤s4Bδ(tan−1(sBδ (x2 −z)) −tan−1(sBδ (x1 −z))) + 1 .Proof:To get the lower estimate, compare ϕ with the time one map of the flowdxdt = 2A(x −z)2 + δ .We claim that iterations of the time one map of this flow overtake iterations of the mapx →x + A(x −z)2 provided that δ is small enough, i.e.if l is sufficiently large.Letx ∈(x1, x2). We want2A(x + A(x −z)2 −z)2 ≥A(x −z)2 .Observe that there is nothing to prove if x > z.

Choose K so that KA < 1/2 to concludethe proof.Then the lower estimate follows by integrating the flow. The upper estimate can bedemonstrated in analogous way.Q.E.D.27

We will assume that the assumption z−x1 < K is always satisfied, perhaps by making Weven smaller. As a corollary to Lemma 2.3, we note that for l large, δ ∼T −2 where the ∼signmeans that the ratio of the two quantities is bounded from both sides by positive constants.Indeed, taking x1 −am and x2 −al−m we see that the arguments of the arctan terms arevery large for large l, meaning that they values are close to π/2, or −π/2 respectively.

Inparticular, δ ∼δ′ for two different almost parabolic maps.Lemma 2.4 For every b > 0 there are constants l0 and β > 0 with the property that if|ap −z| ≤b√δ, thenp −mT> β and l −m −pTβ .Proof:Note that z is separated from the boundary of W by a distance which is at least a positiveconstant. This is because the “step” of the orbit of a is still quite large at the boundary ofW.

When δ is sufficiently small (depending on b) this means that the entire neighborhood(z −b√δ, z + b√δ) is in a positive distance from the boundary of W. This means that forx1 = am and x2 = ap the time of passage is ∼q1/δ. The same reasoning applies to thepassage from ap to the other end of W. The claim follows from Lemma 2.3.Q.E.D.The approximation formula given by Lemma 2.3 works well for points which are withina distance of the order of√δ from 0.

We need a different formula for points further away.Lemma 2.5 Choose a point ap so that the distance from ap to z is bigger than b√δ. Let tdenote p −m if ap < z or l −m −p otherwise.

There are constants l0, b0, C > 0 and c > 0so that if l ≥l0 and b ≥b0, thenc|z −x| ≤t ≤C|z −x| .Proof:We begin by comparing the iterations of ϕ with flows like in the proof of Lemma 2.3. Givena flowdxdt = δ + A(x −z)2we compare it with the flowdxdt = A(x −z)2 .The discrepancy between time t < T maps of these flows is less than Tδ ∼√δ.

So, forb0 large, approximating by the simpler flow modifies |z −x| only by a bounded factor.Integration gives the desired estimate.Q.E.D.Observe that Lemma 2.5 implies a “converse” of Lemma 2.4. Namely,28

Lemma 2.6 Under the hypotheses and using notations of Lemma 2.5, if t/T > β > 0, thenthere is a number q depending on β so that|ap −z| ≤q√δ .Proof:From Lemma 2.5, we getx ∼t−1 ∼β−1T −1 ∼β−1√δ .Q.E.D.Proof of Proposition 2.Let us conclude the proof. If |ap−z| ≤b√δ, then by Lemmas 2.6and 2.4 we get |a′p −z′| ≤b′√δ where b′ depends on b.

Then on the interval (ap, ap+1) thefunction u indeed has a slope bounded from both sides by positive numbers depending on b.This follows from the approximation given by Lemma 2.3 used with x1 := ap and x2 := ap+1or x1 := a′p and x2 := a′p+1 respectively. Since the arguments of the arctan functions arebounded in all cases, the estimate follows.

Because of the symmetry of the problem, onecan also bound the slope of u on (ap, ap+1) assuming that |a′p −z′| ≤b′√δ′. The case whichremains is when both |ap −z| and |a′p −z′| are large compared with δ.

But then Lemma 2.5is applicable showing that |ap −z| ∼|a′p −z′| and since|ap −ap+1| ∼(ap −z)2 ∼(a′p −z′)2 ∼|a′p −a′p+1|the slope is bounded as needed.We finally remark that all was done assuming l sufficiently large. In the case of boundedl Proposition 2 is easy.

So, the proof has been finished.Construction of the branchwise equivalence.Faced with a case of almost paraboliccritical pull-back, we first “mark” the branchwise equivalence so that upon its first exit fromthe central domain the critical value of φ is mapped onto the critical value of ˆφ. This willadd only bounded conformal distortion by Lemma 2.1.

Next, we build a quasisymmetricmap u which transforms B (the box through which the central branch ranges) onto ˆB, andthe backward orbit of the endpoint a of the central domain by the central branch onto thecorresponding orbit in the phase space of ˆφ. The hard part of the proof is that this u isuniformly quasisymmetric, and that follows from Proposition 2.

Then we change u betweenthe boundary of B and the central domain by υ0, and between ai and ai+1, i ≤l −2 byˆφ−i−1 ◦υ0 ◦φi+1 .By Lemma 3.14 of [18], this will give a uniformly quasisymmetric map on J \ (al−2, −al−2)provided that it is quasisymmetric on any interval (ai−1, ai+1). This is clear since we arepulling-back by bounded diffeomorphisms, with the exception of the last interval (al−2, al−2)where this is a quadratic map composed with a diffeomorphism.

Still, it remains quasisym-metric.29

Conclusion of the proof of Proposition 1.To finish the proof of Proposition 1, we stillneed to check that the standard replacement condition is satisfied with uniform norm. Thefact that υ∞restricted to each individual domain replaces υ∞on J is implied by the factthat on each newly created domain υ∞is a lift by diffeomorphism or folding branches, alwaysǫ-extendible, of υ′ from another branch.

The technical details of this fact are provided in [18],Lemma 4.6. Then the fact that υ∞restricted to each box also replaces υ∞from J followsautomatically.

We just notice that boxes of φ∞, with the exception of the central domain,are the same as the boxes of φ. The construction of the Sewing Lemma can be conductedfor Y extending υ0 from one box only, so it will not affect the replacement condition.

Thisconcludes the proof of Proposition 1.Theorem about initial inducing.Definition 2.9 A real box map is called suitable if the critical value of the central branchis in the central domain and stays there forever under iterations of the central branch.The inducing construction described earlier in this section hangs up on a suitable map.In is easy to see that if a map is suitable, there must be an interval symmetric with respectto 0 which is mapped inside itself by the central branch. In fact, this interval must be arestrictive interval of the original f.Theorem 2 Suppose that f and ˆf belong to Fη for some η > 0 and are topologically con-jugate.

Then, for every α > 0, we claim the existence of conjugate box mappings φs and ˆφson the respective fundamental inducing domains J and ˆJ, and a branchwise equivalence υsbetween them so that a number of properties are satisfied.• φs and ˆφs are either full, or of type I (both of the same type. )• φs and ˆφs both possess standard ǫ-extendibility in the sense given by Definition 1.14.• υs satisfies the standard replacement condition with distortion K2 in the sense ofDefinition 2.2.• υs is K1-quasisymmetric.In addition, φs and ˆφs either are both suitable or this set of conditions is satisfied:• for any domain D of φs or ˆφs which does not belong to monotone branch which rangesthrough the fundamental inducing domain,|D|dist(D, ∂J) ≤α ,• if the mappings are of type I, then|B′|dist(B, ∂J) ≤αand the analogous condition holds for ˆφs,30

• they both have extensions as complex box mappings with hole structures satisfying thegeometric bound K3.• on the boundary of each hole, the mapping onto the box is K4- quasisymmetric.The bounds ǫ and Ki depend only on η and α.An outline of the proof of Theorem 2.Theorem 2 is going to be the main resultof this section.The strategy will be to obtain φs and its counterpart in a bounded (interms of η and α) number of basic inducing operations. Then ǫ-extendibility will be satisfiedby construction.

The main difficulty will be to obtain the bounded hole structure. Next,the branchwise equivalence will be constructed based on Proposition 1, so not much extratechnical work will be required other than checking the assumptions on this Proposition.2.2Inducing processThe starting point.Fact 2.5 Under the hypotheses of Theorem 2, conjugate induced mappings φ0 and ˆφ0 existwith a branchwise equivalence υ0.

Also, versions of φ0 and ˆφ0 exist which are refined at theboundary with appropriate branchwise equivalences. In particular, υ0,b is infinitely refined atboth endpoints.

The following conditions are fulfilled for φ0, ˆφ0 and all their versions refinedat the boundary. For simplicity, we state them for φ0 only.• φ0 has standard ǫ-extendibility,• every domain except for two at the endpoints of J is adjacent to two other domainsand the ratio of lengths of any two adjacent domains is bounded by K1,• υ0 restricted to any domain replaces υ0,b with distortion K2 and coincides with υ0,boutside of J,• υ0 is K3-quasisymmetric,• there are two monotone branches which range through J with domains adjacent to theboundary of J, and the ratio of length of any of them to the length of J is at least ǫ1.The estimates ǫ, ǫ1 and Ki depend only on η.Proof:This fact is a restatement of Proposition 1 and Lemma 4.1 of [18].Q.E.D.31

The general inducing step.Suppose that a real box mapping φ is given which is eitherfull, of type I, or of type II and not suitable. Let us also assign a rank to this mapping whichis 0 is the φ is full.

By the assumption that the critical value is recurrent, the critical valueφ(0) is in the domain of φ. There are three main distinctions we make.Close and non-close returns.

If the critical value is in the central domain, the case isclassified as a close return. Otherwise, it is described as non-close.

For a close return, welook at the number of iterations of the central branch needed to push the critical value outof the central domain. We call it the depth of the close return.Box and basic returns.

The situation is described as basic provided that upon its firstexit from the central domain the critical value falls into the domain of a monotone branchwhich ranges through J. Otherwise, we call it a box case.High and low returns.We say that a return is high if 0 is the range of the centralbranch, otherwise the case is classified as low.

Clearly, all eight combinations can occur,which together with three possibilities for φ (full, type I, type II) gives us twenty-four cases.In the description of procedures, we use the notation of Definition 1.9 for boxes of φ.In all cases, we begin with minimal boundary-refinement.A low, close and basic return.In this case the almost parabolic pull-back is exe-cuted. The resulting box mapping is usually not of any classified type.

This is followed bythe filling-in of all branches ranging through B′ or B. This gives a type I mapping.

Finally,the critical pull-back is applied to give us a full map.A low, close and box return.Again, we begin by applying almost parabolic pull-back. Directly afterwards, we apply simple critical pull-back.

The result will always be atype II mapping.A high return, φ full or type I.Critical pull-back with filling-in is applied. Theoutcome is of type I in the box case, and full in the basic case.A high return, φ of type II.Apply filling-in to obtain a type I map, then follow theprevious case.A low return, non-close and basic, φ not of type II.Apply critical pull-back withfilling-in.

The outcome is a full map.A low return, non-close and basic, φ of type II.Use filling-in to pass to type I,then follow the preceding step.A low and box return.Apply simple critical pull-back. The outcome is a type IImap.This is a well-defined algorithm to follow.

The rank of the resulting mapping is incre-mented by 1 in the box case, and reset to 0 in the basic case.32

Notation.A box mapping is of rank n, we will often denote Bn := B and Bn′ := B′.Features of general inducing.Lemma 2.7 If φ is a type I or II induced box mapping of rank n derived from some f ∈Fby a sequence of generalized inducing steps, then|Bn||Bn′| ≤1 −ǫwhere ǫ is a constant depending on η only.Proof:If φ is full, the ratio is indeed bounded away from 1 since a fixed proportion of B0 is occupiedby the domains of two branches with return time 2. In a sequence of box mappings this ratioremains bounded away from 1 by Fact 4.1 of [10].Q.E.D.Lemma 2.8 Let φn denote the mapping obtained from phi0 given by Fact 2.5 in a sequence ofn general inducing steps.

There are sequences of positive estimates ǫ(n) and ǫ1(n) dependingotherwise only on η so that the following holds:• φn has standard ǫ(n)-extendibility,• the margin of extendibility of branches ranging through J remains unchanged in theconstruction,• all branches are ǫ(n)-extendible,• for every box B smaller than J, both external domains exist, belong to monotonebranches mapping onto J, and the ratios of their lengths to the length of B are boundedfrom below by ǫ1(n).Proof:This lemma follows directly from analyzing the multiple cases of the general inducing step.The second claim follows directly since we use minimal boundary refinement. For the firstclaim, in most cases one uses Lemma 1.3.

The more difficult situation is encountered ifa simple critical pull-back occurs. In this situation, the extendibility of short monotonebranches may not be preserved.However, one sees by induction that a short monotonedomain is always adjacent to two long monotone domains.

The ratio of their lengths canbe bounded in terms of η and n. So, if the extendibility of this short monotone branch isdrastically reduced, then the critical value must be in this adjacent long monotone domain.But then a basic return has occurred in which the general inducing step always uses criticalpull-back with filling-in, which preserves extendibility.Q.E.D.33

Main propositionProposition 3 For every α > 0, there are a fixed integer N and β0 > 0, both independentof the dynamics, for which the following holds. Given a mapping φ0 obtained from Fact 2.5there is a sequence of no more than N general inducing steps followed by no more than Nsteps of simultaneous monotone pull-back which give an induced map φ that satisfies at leastone of these conditions:• φ is of type I and suitable,• φ is full and α-fine,• φ is of type I, has a hole structure of a complex box mapping with geometrical normless than β0, and for every domain of a short monotone branch or the central domain,the ratio of its length by the distance from the boundary of I is less than α.Proposition 3 is a major step in proving Theorem 2.

Observe that the first and thirdpossibilities are already acceptable as φs in Theorem 2.Taking care of the basic case.We construct an induced map φ(α) according by applyingthe general inducing step until the central domain and all short monotone domains D satisfy|D|/dist(D, ∂J) ≤α. This will take a bounded number of general inducing steps because thesizes of central domains shrink at a uniform exponential rate (see 2.7.) Of course, we maybe prevented from getting to this stage by encountering a suitable map before; in that caseProposition 3 is already proven.

From the stage of ϕ(δ) on, whenever we hit a basic returnwe are in the second case of Proposition 3. Indeed, the ratio of length of the central domainor any short monotone domain to the distance from the boundary of J is small.

What is leftis only to shorten long monotone, non-external domains. This is done in a bounded numberof simultaneous monotone pull-back steps.

So we can assume that box returns exclusivelyoccur beginning from ϕ(α).2.3Finding a hole structureWe consider the sequence (ϕk) of consecutive box mappings obtained from ϕ0 := ϕ(α)in a sequence of general inducing steps. The objective of this section is to show that forsome k which is bounded independently of everything else in the construction, a uniformlybounded hole structure exists which extends ϕk as a complex box mapping in the sense ofDefinition 1.8.The case of multiple type II maps.We will prove the following lemma:Lemma 2.9 Consider some ϕm of rank n. There is a function k(n) such that if the mappingsϕm+1, .

. .

, ϕm+k(n) are all of type II, then ϕm+k(n) has a hole structure which makes it acomplex box mapping. The geometric norm of this hole structure is bounded depending solelyon n and η.34

Proof:If a sequence of type II mappings occurs, that means that the image of the central branchconsistently fails to cover the critical point. The rank of all branches is fixed and equal ton.

The central domain shrinks at least exponentially fast with steps of the construction ata uniform rate by Lemma 2.7. Thus, the ratio of the length of the central domain of ϕm+kto the length of Bn is bounded by a function of k which also depends on n, and for a fixedn goes to 0 as k goes to infinity.This means that we will be done if we show that a small enough value of this ratio ensuresthe existence of a bounded hole structure.

We choose two symmetrical circular arcs whichintersect the line in the endpoints of Bn at angles π/4 to be the boundary of the box. (α willbe chosen in a moment, right now assume α < π/2).

We take the preimages of the box by themonotone branches of rank n. They are contained in similar circular sectors circumscribedon their domains by Poincar´e metric considerations given in [27]. Now, the range of thecentral branch is not too short compared to the length of Bn since it at least covers onebranch adjacent to the boundary of the box (otherwise we would be in the basic case).

Weclaim that the domain of that external branch constitutes a proportion of the box boundeddepending on the return time of the central branch. Indeed, one first notices that the returntime of the external branch is less than the return time of the central branch.

This followsinductively from the construction. But the range of the external domain is always the wholefundamental inducing domain, so the domain of this branch cannot be too short.

On theother hand, the size of the box is uniformly bounded away from 0. To obtain the preimageof the box by the complex continuation of the central branch, we write the central branchas h(z −1/2)2.

The preimage by h is easy to handle, since it will be contained in a similarcircular sector circumscribed on the real preimage. Since the distortion of h is again boundedin terms of n, the real range of (x−1/2)2 on the central branch will cover a proportion of theentire h−1(Bn) which is bounded away from 0 uniformly in terms of n. Thus, the preimageof the complex box by the central branch will be contained in a star-shaped region whichmeets the real line at the angle of π/4 and is contained in a rectangle built on the centraldomain of modulus bounded in terms of n.It follows that this preimage will be contained below in the complex box if the middle do-main is sufficiently small.

As far as the geometric norm is concerned, elementary geometricalconsiderations show that it is bounded in terms of n.Q.E.D.Formation of type I mappings.We consider then same sequence ϕk and we now analyzethe cases when the image of the central branch covers the critical point. Those are exactlythe situations which lead to type I maps in general inducing with filling-in.A tool for constructing complex box mappings.The reader is warned that thenotations used in this technical fragment are “local” and should not be confused with symbolshaving fixed meaning in the rest of the paper.

The construction of complex box mappings(choice of a bounded hole structure ) in the remaining cases will be based on the technicalwork of [20]. We begin with a lemma which appears there without proof.35

Lemma 2.10 Consider a quadratic polynomial ψ normalized so that ψ(1) = ψ(−1) = −1,ψ′(0) = 0 and ψ(0) = a ∈(−1, 1). The claim is that if a < 12, then ψ−1(D(0, 1)) is strictlyconvex.Proof:This is an elementary, but somewhat complicated computation.

We will use an analyticapproach by proving that the image of the tangent line to ∂ψ−1(D(0, 1)) at any point islocally strictly outside of D(0, 1) except for the point of tangency. We represent points inD(0, 1) in polar coordinates (r, φ) centered at a so that φ(1) = 0, while for points in preimagewe will use similar polar coordinates r′, φ′.

By school geometry we find that the boundaryof D(0, 1) is given by(r + a cos φ)2 + a2 sin2 φ = 1 .By symmetry, we restrict our considerations to φ ∈[0, π]. We can then change the parameterto t = −a cos φ, which allows us to express r as a function of t for boundary points, namelyr(t) =√1 −a2 + t2 + t .

(3)Now consider the tangent line at the preimage of (r(t), t) by ψ. By conformality, it isperpendicular to the radius joining to 0, so it can be represented as the set of points (r′, φ′)r′ =qr(t)cos2(θ/2) , φ′ = π2 −φ2 + θ2where θ ranges from −π to π.

The image of this line is given by (ˆr(θ), φ −θ) whereˆr(θ) =2r(t)1 + cos θ .Let us introduce a new variable t(θ) := −a cos(φ −θ) so that t = t(0). Our task is to provethatr(t(θ)) < ˆr(θ)(4)for values of θ is some punctured neighborhood of 0.

This will be achieved by comparing thesecond derivatives with respect to θ at θ = 0. By the formula [3]r(t(θ)) =q1 −a2 + t(θ)2 + t(θ) .The second derivative at θ = 0 is1 −a2(q1 −a2 + t(θ)2)3(a2 −t2) −t −t2q1 −a2 + t(θ)2 == −q1 −a2 + t(θ)2 −t +1 −a2(q1 −a2 + t(θ)2)3 .The second derivative of the right-hand side of the desirable inequality [4] is more easilycomputed asq1 −a2 + t(θ)2 + t2.36

Thus, the proof of the estimate [4], as well as the entire lemma, requires showing that32(q1 −a2 + t(θ)2 + t) −1 −a2(q1 −a2 + t(θ)2)3 > 0for |a| < 1/2 and |t| ≤|a|. For a fixed a, the value of this expression increases with t. So,we only check t = −a which reduces to32(1 −a) −(1 −a2) > 0which indeed is positive except when a ∈[1/2, 1].Q.E.D.The main lemma.Now we make preparations to prove another lemma, which isessentially Lemma 8.2 of [20].Consider three nested intervals I1 ⊂I0 ⊂I−1 with thecommon midpoint at 1/2.Suppose that a map ψ is defined on I1 which has the formh(x −1/2)2 where h is a polynomial diffeomorphism onto I−1 with non-positive Schwarzianderivative.

We can think ψ as the central branch of a generalized box mapping. We denoteα := |I1||I0|.

Next, if 0 < θ ≤π/2 we define D(θ) to be the union of two regions symmetrical withrespect to the real axis. The upper region is defined as the intersection of the upper halfplane with the disk centered in the lower ℜ= 1/2 axis so that its boundary crosses the realline at the endpoints of I0 making angles θ with the line.

So, D(π/2) is the disk having I0as diameter.Lemma 2.11 In notations introduced above, if the following conditions are satisfied:• ψ maps the boundary of I−1 into the boundary of I0,• the image of the central branch contains the critical point,• the critical value inside I0, but not inside I1,• the distance from the critical value to the boundary of I0 is no more than the (Haus-dorff) distance between I−1 and I0,then ψ−1(D(θ)) is contained in D(π/2) and the vertical strip based on I1. Furthermore, forevery α < 1 there is a choice of 0 < θ(α) < π/2 so thatψ−1(D(θ(α))) ⊂D(θ(α))with a modulus at least K(α), and ψ−1(D(θ(α))) is contained in the intersection of twoconvex angles with vertices at the endpoints of I1 both with measures less than π −K(α).Here, K(α) is a continuous positive function.37

Proof:By symmetry, we can assume that the critical value, denoted here by c, is on the left of 1/2.Then t denotes the right endpoint of I0, and t′ is the other endpoint of I0. Furthermore, xmeans the right endpoint of Bn−1.

By assumption, h extends to the range (t′, x). To get theinformation about the preimages of points t, t′, c, x one considers their cross-ratioC = (x −t)(c −t′)(x −c)(t −t′) ≥1 + α4where we used the assumption about the position of the critical value relative I0 and I−1.The cross ratio will not be decreased by h−1.

In addition, one knows that h−1 will map thedisk of diameter I0 inside the disk of diameter h−1(Bn) by the Poincar´e metric argumentof [27]. As a consequence of the non-contracting property of the cross-ratio, we geth−1(c) −h−1(t′)h−1(t) −h−1(t′) < 1 + α4.

(5)When we pull back the disk based on h−1(I0), we will get a figure which intersects the realaxis along I1. Notice that by the estimate [5] and Lemma 2.10, the preimage will be convex,thus necessarily contained in the vertical strip based on I1.Its height in the imaginarydirection is|I1|2vuut h−1(t) −h−1(c)h−1(c) −h−1(t′) < |I1|2s3 −α1 + α ,(6)where we used the estimate [5] in the last inequality.

Clearly,ψ−1(D(π/2))is contained in the disk of this radius centered at 1/2. To prove thatψ−1(D(π/2)) ⊂D(π/2) ,in view of the relation [6] we needαs3 −α1 + α < 1(7)By calculus one readily checks that this indeed is the case when α < 1.To prove theuniformity statements, we first observe thatψ−1(D(θ)) ⊂ψ−1(D(π/2))for every θ < pi/2.

Since [7] is a sharp inequality, for every α < 1 there is some range ofvalues of θ below π/2 for which ψ−1(D(θ)) ⊂D(θ) with some space in between. We onlyneed to check the existence of the angular sectors.

For the intersection of ψ−1(D(θ)) with anarrow strip around the real axis, such sectors will exist, since the boundary intersects thereal line at angles θ and is uniformly smooth. Outside of this narrow strip, even ψ−1(D(π/2))is contained in some angular sector by its strict convexity.Q.E.D.The assumption of extendibility to the next larger box is always satisfied in our construc-tion.38

The case when there is no close return.We now return to our construction andusual notations. We consider a map ϕk, type II and of rank n, whose central branch coversthe critical point, but without a close return.

Then:Lemma 2.12 Either the Hausdorffdistance from Bn to Bn−1 exceeds the Hausdorffdistancefrom Bn−1 to Bn−2, or ϕk has a hole structure uniformly bounded in terms of n.Proof:Suppose the condition on the Hausdorffdistances fails. We choose the box around Bn andthe hole around Bn+1 by Lemma 2.11.Observe that the quantity α which plays a rolein that Lemma is bounded away from 1 by Lemma 2.7.

The box is then pulled back bythese monotone branches and its preimages are inside similar figures built on the domainsof branches by the usual Poincar´e metric argument of [27]. For those monotone branches,the desired bounds follow immediately.Q.E.D.Proof of Proposition 3.This is just a summary of the work done in this section.

Weclaim that we have proved that either a map with a box structure can be obtained fromϕ(δ) in a uniformly bounded number of steps of general inducing, or the Proposition 3 holdsanyway. Since Lemmas 2.9 and 2.12 provide uniform bounds for the hole structures in termsof k or the rank which is bounded in terms of k, it follows that the hole structure is boundedor the starting condition holds anyway.

If the inducing fails within this bounded number ofsteps because of a suitable map being reached, then the stopping time on the central branchof the suitable map is bounded, hence Proposition 3 again follows.So, we need to prove that claim. If the claim fails, then by Lemma 2.9 the situations inwhich the image of the central branch covers the critical point have to occur with definitefrequency.That is, we can pick a function m(k) independent of other elements of theconstruction which goes to infinity with k such that among ϕ1, .

. .

, ϕk the situation in whichthe critical point is covered by the image of the central branch occurs at least m(k) times.But each time that happens, we are able to conclude by Lemma 2.12 that the Hausdorffdistance between more deeply nested boxes is more than between shallower boxes. Initially,for ϕ0 whose rank was n, the Bn distance between and Bn−1 was a fixed proportion of thediameter of Bn.

So only a bounded number of boxes can be nested inside Bn−1 with fixedspace between any two of them. So we have a bound on the value of m(k), thus on k. Thisproof of the claim is a generalization of the reasoning used in [20].

The claim concludes theproof of Proposition 3.2.4Proof of Theorem 2Immediate cases of Theorem 2.Given two conjugate mappings φ0 and ˆφ0 obtainedby Fact 2.5, we apply the generalized inducing process to both. By Proposition 3 after abounded number of generalized inducing steps we get to one of the three possibilities listedthere: a suitable map, a full map, or a type I complex box map with a bounded hole structure.In the first and third cases all we need in order to conclude the proof is the existence of a39

branchwise equivalence υs with needed properties. In the second case extra work will berequired.We will show that generally after n general inducing steps stair from a pair of conjugateφ0 and ˆφ0 we get a branchwise equivalence υ which satisfies• υ coincides with υ0,b outside of J,• υ satisfies the standard replacement condition with distortion K2(n) in the sense ofDefinition 2.2.• υs is K1(n)-quasisymmetric.The bounds K1(n) and K2(n) depend only on η and n.This follows by induction with respect to n from Proposition 1 using Lemma 2.8.

Thismeans Theorem 2 has been proved except in the second case of Proposition 3.In theremaining case, we still have the branchwise equivalence with all needed properties.The case of φ full.We need to construct a hole structure. Observe that we can assumethat the range of the central branch of φ is not contained in an external branch.

In thatcase we could compose the central branch with this external branch until the critical valueleaves the external domain.This would not change the branchwise equivalence, box orextendibility structures of the map. Also, instead of φ rather consider the version φb refinedat the endpoint not in the range of the central domain enough times to make all domainsinside this external domain of φ shorter than some α′.A full map with two basic returns.Let us first assume that φb shows a basicreturn.

Carry out a general inducing step. The resulting full mapping φ1 has short monotonebranches which are ǫ′-extendible and ǫ′ goes to 1 as α and α′ go to 0.

Indeed, these shortbranches extend with the margin equal to the central central domain of φ, and if α is smallthis is much larger than the central domain of φ1. Let δ denote the distance from the criticalvalue of φ1 to the boundary of J.

We claim that for very δ0 > 0 there is an α > 0, otherwiseonly depending on η, so that if φ was α-fine, then a bounded hole structure exists. Indeed,take the diamond neighborhood with height β of J as the complex box.

Its preimage by thecentral branch is quasidisc with norm depending of δ, β and the extendibility (thus ultimatelyon η.) If β is very small, the preimage is close to the “cross” which is the preimage of Jby the central branch is the complex plane.

In particular, for β small enough it fits insidea rombe symmetrical with respect to the real axis with the central domain as a diagonal.The diameter of this rombe is bounded in terms of δ. If the extendibility of short monotonebranch is sufficiently good, then the preimages of this rombe by the short monotone brancheswill be contained in similar rombes around short monotone domains (see Fact 2.1 and useK¨obe’s distortion lemma.) This gives us a bounded hole structure.Finally, we show how to modify φ1 so that its critical value is in a definite distance fromthe boundary of J.

If the range of the central branch is very small, this is very easy. Justcompose the central branch with the external branch a number of times to repel the criticalvalue from the endpoint, but so that it is still inside the external domain, thus giving a basic40

return. This may require an unbounded number of compositions, but they will not changethe branchwise equivalence.

If the range of the central branch is almost the entire J, choosea version of φ1 which is refined to the appropriate depth at the endpoint of J not in therange of the central branch. The appropriate depth should be chosen so that the image ofthe critical value by the external branch of the refined version is still in an external domain,but already in a definite distance from the boundary.

The possibility of doing this followssince we can bound the eigenvalue of the repelling periodic point in the boundary of J fromboth sides depending on η (see Fact 2.3 in [18].) Then apply the general inducing step tothis version of φ1, and the construct the hole structure as indicated above for the resultingmap.Since in this process we only use a bounded number of inducing operations, or operationsthat do not change the branchwise equivalence, the branchwise equivalence between the mapsfor which we constructed hole structures will satisfy the requirements of Theorem 2.

So, inthe case of a double basic return we finished the proof.A box return.In this case, we apply the general inducing step once to get a type IImapping φ1. Observe that the central domain of φ1 is very short compared to the size ofthe box (the ratio goes to 0 with α, independently of α′. ) Again, if the critical value is ina definite distance from the boundary of the box, compared with the size of the box, thenwe can repeat the argument of Lemma 2.9 to build a hole structure.

Otherwise, the criticalvalue is in a long monotone branch external in the box. So one more application of thegeneral inducing step will give us a mapping which is twice basic, so we apply the previousstep.

Again, we see that the branchwise equivalence satisfies the requirements of Theorem2. This means that Theorem 2 has been proved in all cases.3Complex pull-back3.1IntroductionWe will introduce a powerful tool for constructing branchwise equivalences while preservingtheir quasiconformal norm.

Since the work is done in terms of complex box mappings, wehave to begin by defining an inducing process on complex box mappings.Complex inducing.A simple complex inducing step.Suppose that φ is a complex box mapping whichis either full or of type I. We will define an inducing step for φ which is the same as thegeneral inducing used in the previous section on the real, with the only difference that infiniteboundary-refinement is used.

First, perform the infinite boundary refinement. This has anobvious meaning for complex box mappings.

Namely, one finds bad long monotone branchesand composes their analytic continuations with φr. Call the resulting map φ′.

Next, replacethe central branch of φ′ with the identity to get φr. Next, construct ˜φ which is the same as41

φ outside of the central domain, and put˜φ := φr ◦φon the central domain. Finally, fill in all short univalent branches of ˜φ to get a type I or fullmapping.

Observe that on the real line this is just boundary refinement followed by criticalpull-back with filling-in, so standard extendibility is preserved.We also distinguish complex inducing without boundary-refinement. This is the same asthe procedure described above only without boundary refinement.A complete complex inducing step.A complete complex inducing step, which willalso be called a complex inducing step is the same as the simple inducing step just definedprovided that φ does not show a close return, that is, the critical value of φ is not in thecentral domain.

If a close return occurs, a complete inducing step is a sequence of simpleinducing steps until a mapping is obtained which shows a non-close return. Then the simpleinducing step is done once again and the whole procedure gives a complete inducing step.If simple complex inducing steps are used without boundary refinement, we talk about acomplex inducing step without boundary refinement.External marking.Consider two equivalent conjugate complex box mappings, ϕ and ˆϕ,see Definition 1.8 of complex box mappings.Definition 3.1 A quasiconformal homeomorphism Υ is called an externally marked branch-wise equivalence if it satisfies this list of conditions:• restricted to the real line, Υ is a branchwise equivalence in the sense of Definition 1.4,• Υ maps each box of ϕ onto the corresponding box of ˆϕ,• Υ maps each complex domain of ϕ onto a complex domain of ˆϕ so that these domainsrange through corresponding boxes,• On the union of boundaries of all holes complex domains of ϕ, the functional equationˆϕ ◦Υ = Υ ◦ϕholds.The last condition of this definition will be referred to as the external marking conditionby analogy to internal marking which will be introduced next.Internal marking.Definition 3.2 Let υ be a branchwise equivalence.

An internal marking condition is definedto a choice of a set S so that S is contained in the union of monotone rank 0 domains of υand each such domain contains no more than one point of S. The branchwise equivalence υwill be said to satisfy the internal marking condition if υ coincides with the conjugacy on S.42

Definition 3.3 We will say that a branchwise equivalence Υ is completely internally markedif for each internal marking condition Υ can be modified without changing it on the boundariesof holes and diamonds so that the marking condition is satisfied. We will say that an estimate,for example a bound on the quasiconformal norm, is satisfied for a fully internally markedΥ, if all those modifications can be constructed so as to satisfy this estimate.Definition 3.4 Two generalized induced mappings φ and ˆφ are called equivalent providedthat a branchwise equivalence υ exists between them such that for every domain D of φ, wehaveˆφ(υ(D)) = υ(φ(D)) .Equivalence of induced mappings is weaker than their topological conjugacy and it merelymeans that the branchwise equivalence respects the box structures, that correspondingbranch range through corresponding boxes, and that the critical values are in correspondingdomains.We are ready to state out main result.Theorem 3 Suppose that φ and ˆφ are conjugate complex box mappings with diamonds oftype I or full.

Suppose that Υ is Q-quasiconformal externally marked and completely inter-nally marked branchwise equivalence. Also, a branchwise equivalence Υb is given between theversions of φ and ˆφ infinitely refined at the boundary.

The map Υb is also Q-quasiconformal,externally marked and completely internally marked and coincides with Υ on the boundariesof all boxes. Suppose also that Υ or Υb restricted to the domain of any branch that rangesthrough J replaces Υb on J with distortion K′ and suppose φ and ˆφ satisfy standard ǫ-extendibility.

Suppose that φ1 and ˆφ1 are obtained from φ and ˆφ respectively after severalcomplex inducing steps, some perhaps without boundary refinement (but always the sameprocedure is used on both conjugate mappings.) Then, there is an externally marked andcompletely internally quasiconformal branchwise equivalence Υ1 between φ1 and ˆφ1.

Further-more, if φ1 is full, then Υ1 is Q-quasiconformal. Otherwise, Υ is Q-quasiconformal on thecomplement of the central domain and the union of short univalent domains.

In boundaryrefinement is used at each step, then φ1 and ˆφ1 still have standard ǫ-extendibility and Υ1restricted to any domain of a branch that ranges through J replaces Υ1 on J with distortionK. The number K only depends on ǫ and K′.

If the construction without boundary refine-ment is used and only the box case occurs, the theorem remains valid if φ and ˆφ are complexbox mappings without diamonds.3.2Proof of Theorem 3.Complex pull-back.Historical remarks.The line of “complex pull-back arguments” initiated by [6] andused by numerous authors ever since. We only mention the works which directly precededand inspired our construction.In [5] the idea of complex pull-back was applied to thesituation with multiple domains and images, all resulting from a single complex dynamicalsystem.

According to [14], a similar approach was subsequently used in [28] in the proof43

of the uniqueness theorem for non-renormalizable quadratic polynomials. Also, [20] useda complex pull-back construction to study metric properties of the so called “Fibonacciunimodal map”.Description.Given φ and ˆφ full or of type I with an externally marked branchwiseequivalence Υ, we will show how to build a branchwise equivalence between φ1 and ˆφ1obtained in just one complex inducing step.

The first stage is boundary refinement. Corre-spondingly, on the domain of each complex branch ζ being refined, we replace Υ withˆζ−1 ◦Υb ◦ζwhere Υb is the branchwise equivalence between the versions refined at the boundary whichcoincides with Υ on the boundaries of boxes.

The map obtained in this way is called Υ′.Next comes the “critical pull-back” stage. We replace Υ on the central domain with thelift to branched coversˆψ−1 ◦Υ′ ◦ψwhere ψ and ˆψ are central branches.

For this to be well-defined, we need Υ(ψ(0)) = ˆψ(0).If this is a basic return, this condition may be assumed to be satisfied because of internalmarking. In the box case, we have to modify Υ inside the complex branch which containsthe critical value.

We can do this modification in any way which gives a quasiconformalmapping and leaves Υ unchanged outside of this complex domain. We also choose the liftwhich is orientation-preserving on the real line.

Call the resulting map ˜Υ.Finally, there is the infinite filling-in. This is realized as a limit process.

Denote Υ0 := ˜Υ.Then Υi+1 is obtained from ˜Υ by replacing it on the domain of each short univalent branchζ with ˆζ−1 ◦Υi ◦ζ. Then one proceeds the limit almost everywhere in the sense of measure.Note that the branchwise equivalence Υ1,b between versions of φ1 and ˆφ1 infinitely refinedat the boundary can be obtained in the same way using φb and ˆφb instead of φ and ˆφrespectively.Also observe that this procedure automatically gives an externally markedbranchwise equivalence, and Υ1 and Υ1,b are equal on the boundaries of all boxes.Complex pull-back on fully internally marked maps.We observe that the con-struction of complex pull-back we just described is well defined not only on individual branch-wise equivalences with holes, but also on families of fully internally marked branchwiseequivalences.

This follows from the recursive nature of the construction. Suppose that ζ,the dynamics inside a hole or a diamond is used to pull back a branchwise equivalence Υ. Wewill show that any marking condition on newly created long monotone domains can be sat-isfied by choosing Υ appropriately marked.

Indeed, long monotone branches of the inducedmap which arises in this pull-back step are preimages of long monotone branches of the mapunderlying Υ. Thus, if s ∈S, one should impose the condition ζ(s) in the image.

This canonly lead to some ambiguity if ζ is 2-to-1. In this case ζ is an univalent map H followed bya quadratic polynomial.

One prepares two versions of marking which are identical on thepart of the real line not in the real image of ζ, and pulls back both of them by H. Whenthey are finally pull back by the quadratic map, they will be the same on the vertical linethrough the critical point of ζ as a consequence of their being equal on the part of real axisnot in the real image. So one can then match the two versions along this vertical line.44

Induction.Now Theorem 3 is proved by induction with respect to the number of complexinducing steps. For one step, we just proved it.

In the general step of induction, the onlythis which is not obvious is why a Q-quasiconformal branchwise equivalence is regained aftera basic return even though the branchwise equivalence on the previous stage was not Q-quasiconformal on short monotone domains and the central domain. To explain this point,use the same notation as in the description of complex pull-back.

The mapping ˜Υ is Q-quasiconformal except on the union of domains of short monotone branches. The key pointis to notice that it is Q-quasiconformal on the central domain, because this is a preimage ofa long monotone domain (we assume a basic return!).

Then for Υi the unbounded conformaldistortion is supported on the set of points which stay inside short monotone branches forat least i iterations. The intersection of these sets has measure 0, which is very easy to seesince each short monotone branch is an expanding in the Poincar´e metric of the box.

So,Υi form a sequence of quasiconformal mappings which converge to a quasiconformal limitand their conformal distortions converge almost everywhere. This limit almost everywhereis bounded by Q.

By classical theorems about convergence of quasiconformal mappings,see [19], the limit of conformal conformal distortions equal to the conformal distortion of thelimit mappings. Thus, the limit is indeed Q-quasiconformal.To finish the proof of Theorem 3 we have to check extendibility and the replacementcondition in the case when boundary refinement is being used.

Extendibility follows directlyfrom Lemma 1.3. Then, Υ1 restricted to any domain whose branch ranges through J isa pull-back by an extendible monotone or folding branch of Υ (or Υb) from another suchdomain.

The replacement condition is then satisfied, which follows from Lemma 4.6 of [18].3.3The box caseBox inducing.Suppose that a complex box mapping φ is given which is either full or oftype I and shows a box return. We then follow the complex inducing step without boundaryrefinement.

This will be referred to as box inducing and is the same as the box inducing usedin [10]. This certainly works on complex box mappings without diamonds.Complex moduli.Given a complex box mapping φ of type I or full, consider the annulusbetween the boundary of of B′ and the boundary of B. Denote its modulus v(φ).

By ourdefinition of box mappings, v(φ) is always positive and finite.Theorem 4 Let φ be complex box mapping of type II. Suppose that φ has a hole structurewith a geometric bound not exceeding K. Let φ0 denote the type I mapping obtained fromφ by filling-in.

Assume that box inducing can be applied to φ0 n times, giving a sequencemaps φi, i = 0, · · ·, n. Then there is number C > 0 only depending on the bound of the holestructure so thatv(φi) ≥Cifor every i.Related results.We state the related results which will be used in the proof. Thisdoes not exhaust the list of related results in earlier papers, see “historical comments” below.The first one will be called “the starting condition”.45

Definition 3.5 We say that a type I or type II box mapping of rank n satisfies the startingcondition with norm δ provided that |Bn|/|Bn′| < δ and if D is a short monotone domain ofφ, then also |D|/dist(D, ∂Bn′) < δ.Fact 3.1 Let φ0 be a real box mapping, either of type I of rank n or full. Pick 0 < τ < 1 andassume that the central branch is τ-extendible.

For every τ, there is a positive number δ(τ)with the following property. Suppose that φi, i ≥0 is sequence of type I real box mappingssuch that φj+1 arises from φj by a box inducing step.

Let φ0 satisfy the starting conditionwith norm δ(τ). Then,|Bn+i|/|B(n+i)′| ≤Ciwhere C is an absolute constant less than 1.Proof:This is Fact 2.2 of [10].

A very similar statement, but for a slightly different inducing processis Proposition 1 of [17].Q.E.D.Fact 3.2 Let φ0 be a complex box mapping. Let φ0, φ1, · · · , φn be the sequence in which thenext map is derived from the preceding one by a box inducing step.

Let Bi denote the centraldomain of φi on the real line, and let Bn′ denote the box on the real line through which thecentral branch ranges. Suppose that for some C < 1 and every i|Bn||Bn′| ≤Ciand suppose that the hole structure of φ0 satisfies a geometric bound β.

For every C < 1 andβ there is a positive C so thatv(φi) ≤Cifor every i.Proof:This is Theorem D of [10].Q.E.D.Historical comments.Theorem 4, in the way we state it, follows from Theorems Band D of [10]. However, we give a different proof based on Theorem D, but not on TheoremB.

Weaker results saying that |Bi|/|Bi′| decrease exponentially fast were proved in variouscases in [17], [20] and later papers, including an early preprint of this work. In the Appendixwe show a result similar to Fact 3.2.

Even though this result is not a necessary step in theproof of the Main Theorem, we think that it may be of independent interest and also itsproof shows the main idea of the proof of Fact 3.246

Artificial maps.Introduction.Artificial maps were introduced in [20]. In our language, [20] showedhow to prove that the starting condition must be satisfied at some stage of the box con-struction for a concrete “Fibonacci” unimodal map.

2 The idea was to use an artificial map“conjugated” to the box map obtained as the result of inducing. Clearly, an artificial mapcan be set up so as to satisfy the starting condition.

Next, one shows that the induced mapand its artificial counterpart are quasisymmetrically conjugate. Since for the artificial mapthe starting condition is satisfied with progressively better norms, after a few box steps itwill be forced upon the induced map by the quasisymmetric conjugacy.This strategy has a much wider range of applicability than the Fibonacci polynomial andit is at the core of our proof of Theorem 4.Topological conjugacy.Lemma 3.1 Given a box mapping induced by some f from F and any homeomorphism ofthe line into itself, a box map (artificial) can be constructed that is topologically conjugateto the original one and whose branches are all either affine or quadratic and folding.

Thehomeomorphism then becomes a branchwise equivalence.Proof:The domains and boxes of the artificial map that we want to construct are given by theimages in the homeomorphism. Make all monotone branches affine and the central branchquadratic.

We show that by manipulating the critical value we can make these box mapsequivalent. We prove by induction that if two maps are similar, by just changing the criticalvalue of one of them we can make them equivalent.

The induction proceeds with respectto the number of steps needed to achieve the suitable map. The initial step is clearly trueby the continuity of the kneading sequence in the C1 topology, and the the induction stepconsists in the remark that by manipulating the critical value in similar maps we can ensurethat they remain similar after the next inducing and the critical value in the next inducedmap can be placed arbitrarily.Q.E.D.The construction of an original branchwise equivalence.We will prove the followinglemma:Lemma 3.2 Under the assumptions of Theorem 4 and for every δ > 0 there is an artificialmap Φ with all branches either affine or quadratic and folding which is conjugate to φ. Next,Φ also has a hole structure which makes it a complex box mapping of type II.

Also, the typeI mapping obtained from Φ by filling-in satisfies the starting condition with norm δ. Also,we claim that an externally marked branchwise equivalence exists between φ and ϕ which isQ quasiconformal. The number Q only depends on δ and the geometric norm of the holestructure for φ (K in Theorem 4).2The Fibonacci map is persistently recurrent in the sense of [28], or of infinite box type in the senseof [17].47

Proof:First construct the artificial map Φ. To this end, we choose a diffeomorphism h which isthe identity outside B′ and squeezes the central branch so that the after filling-in the type Imap satisfies the starting condition.

We observe that the “nonlinearity” h′′/h′ can be madebounded in terms of the bound on the hole structure. Then by Lemma 3.1 we can adjustthe critical value of this artificial map, without moving domains of branches around, so thatthe map is equivalent to φ.

This defines Φ.Since Φ has affine or quadratic branches, the existence of a bounded hole structureequivalent to the structure already existing is clear. One simply repeats the arguments ofLemmas 2.12 and 2.9 with “infinite extendibility” which makes the problem trivial.The final step is to construct the branchwise equivalence by Lemma 1.2.

First, we decidethat on the real line the branchwise equivalence is h. We next define Υ inside the complexbox around B′. On the boundary of the box, we just take a bounded quasiconformal map.Again by K¨obe’s lemma this propagates to the holes with bounded distortion.

Since h is adiffeomorphism of bounded nonlinearity inside each hole, it can be filled with a uniformlyquasiconformal mapping. Lemma 1.2 works to build Υ inside the complex box belonging toB′ with desired uniformity.

Then the map is extended outside of the box. This can also beachieved by Lemma 1.2 regarding B′ as a tooth, holes outside of the box as other teeth, andchoosing a big mouth.Q.E.D.3.4Marking in the box caseIn the box case Theorem 3 does not imply that quasiconformal norms stay bounded.

Onthe other hand, internal marking does not work either in the box case. We should a specialprocedure of achieving an internal marking condition.We remind the reader that v(φ)denotes the modulus between B′ and B when φ is of type I, or between B0 and B when φis full.Proposition 4 Let ˜φ and ˜Φ be conjugate complex box mappings either full or of type I,not suitable and showing a box return.

Suppose that φ and Φ are derived from ˜φ and ˜Φrespectively, and not suitable and show a box return. Let Υ be a Q-quasiconformal branchwiseequivalence acting into the phase space of Φ.

Let v denote the minimum of v(˜Φ) and v(Φ).Then there is a branchwise equivalence Υ′ with the following properties:• Υ′ equals Υ except on the complex domain of φ which contains the critical value of φ,• Υ(φ(0)) = Φ(0),• the quasiconformal norm of Υ′ is bounded byQ + K1 exp(−K2vwhere K1 and K2 are positive constants.Note that the estimate of the quasiconformal norm is independent of the geometry of φ.48

An auxiliary lemma.Lemma 3.3 Let Φ and ˜Φ and v be as in the statement of Proposition 4. Let B0 ⊃B′ ⊃B bethe box structure of Φ.

Choose a complex domain b ∈B′ of Φ which is either short monotone.There exists an annulus A and a constant 0 < C < 1 with the following properties:• A surrounds b and is contained in B′.• the modulus of A is at least Cv and the modulus of the annulus separating A from theboundary of B′ is at least Cv,• the boundary of A intersects the real line at four points, and these points are topologi-cally determined, meaning that if A′ is constructed for a mapping Φ′ conjugated to Φ,the conjugacy will map these four points onto the four points of intersection betweenA′ and the real line.Proof:We split the proof in two cases depending on whether ˜Φ showed a close return or not. Inthe case of a non-close return, the outer boundary of A is chosen as the boundary of thedomain of the analytic extension of the branch from b onto the range B′.

This means that theannulus of A is equal to v(Φ). On the other hand, the domain of this extension is containedin the preimage by the central branch of ˜Φ of some domain of ˜Φ.

A lower bound by v(˜Φ)/2is evident. Also, the topological character of this construction is clear.In the case of a close return, however, this construction would not work because theannulus between the extension domain and the boundary of B′ may become arbitrarilysmall.

So, let us consider the central domain of Φ. Recall that the box inducing step in thecase of a close return is a sequence of simple box inducing steps with close returns ended by asimple box inducing step with a non-close return.

Let Φ1 mean the mapping obtained for ˜Φby this sequence of simple box inducing steps with close returns. The central branch of Φ isthe composition of a restriction of ψ1 (the central branch of Φ1) with a short univalent branchb1 of Φ1.

By construction, this short univalent branch has an analytic continuation whosedomain is contained in the range of ψ1 and whose range is the range of the central branch of˜Φ. Inside this extension domain, there a smaller domain δ mapped onto the central domainof Φ only.

Consider the annulus W between B and the boundary of ψ−1(δ). The modulusof W is a half of the modulus of the annulus between b1 and the boundary of δ, which isv(˜Φ) · 2−l where l is the number of subsequent iterations of ψ which keep the critical valueinside the central domain.

It follows that the modulus of W is at least v(˜Φ)/4. Next, look atthe annulus W ′ separating W from the boundary of B′.

This is at least a half of the annulusseparating δ from the larger extension domain (onto the range of ψ.) This last annulus hasmodulus v(˜Φ).

So W ′ has modulus at least v(˜Φ)/2. Now, to pick A take the extension ofthe branch from b onto B′ and define A to be the preimage of W by this extension.

Since Ais separated from the boundary of the extension domain, let alone from the boundary of B′,by the preimage of W ′, the estimate claimed by this Lemma follows. Also, the topologicalcharacter of the intersection of A with the real line is clear.Q.E.D.49

Proof of Proposition 4.Suppose that the first exit time of the critical value from thecentral domain under iteration by the central branch is l. Call C = Φl(0) and c = φl(0).Suppose C belongs to a short monotone domain b.The point Υ(c) must also be in b. Take the annulus A found for b from Lemma 3.3.Also, call A′ the annulus separating A from the boundary of B′.

Perturb Υ to Υ1 so thatΥ1(c) = C and Υ = Υ1 outside of Υ−1(A). We accept as obvious that this can be done bycomposing Υ with a mapping whose conformal distortion is bounded by k1 exp(−k2mod A).By Lemma 3.3, note that this is a correction allowed by Proposition 4.

Then, we applycomplex pull-back by the central branch ψ to Υ1 l-times.Call the resulting branchwiseequivalence Υ2. If l = 1 it is clear that the conformal distortion of Υ2 is the same as forΥ1.

If l > 1 it less clear since we will have to adjust the branchwise equivalence l −1 timesinside shrinking preimages of B to make c and C correspond. So until just before the lastsimple box inducing step the conformal distortion is not well-controlled.

However, we showas in the proof of Theorem 3 that in this last simple box inducing step, the region where thedistortion was unbounded has preimage of measure 0 because of filling-in, so that ultimatelythe distortion of Υ2 is the same as for Υ1 almost everywhere.Next, construct Υ3 which is the same as Υ outside of Υ−1(b) and equals Φ−1 ◦Υ2 ◦φon b. Both mappings match continuously because of external marking.

Inside b, now lookat A3 = Φ−l−1(A) and A′3 = Φ−l−1(A′). By construction, the mapping Φl+1 in the regionencompassed by these annuli is a branched cover of degree 2, so mod A3 = 12mod A andmod A′3 = 12mod A′.

We consider two cases. If C belongs to the region encompassed bythe outer boundary of A, then Υ3(c) also belongs to this region.

This is because the pointsof intersection of A, and therefore of A3, with the real line are topologically determined byLemma 3.3. So, we perturb Υ3 to leave it unchanged outside of Υ−1(b) and to make thecritical values correspond.

Like in the previous paragraph, we claim that this adjustmentwill only add k1 exp(−k2mod A′3) to the conformal distortion. In this case, we are done withthe proof of Proposition 4.

Otherwise, C belong to the preimage of some short univalentdomain b′ of Φ by Φl+1. Since b′ is nested inside B′ with a modulus at least v(Φ), then thispreimage is nested inside b with a modulus at least half that large.

Also, Υ3(c) must belongto the same preimage of b′. Again, we adjust Υ3 to make the critical values correspond andare done with the proof.Proof of Theorem 4.Given φ from Theorem 4, construct a conjugate artificial mappingΦ from Lemma 4.2 choosing δ from Fact 3.1 for a large τ = 1001 (the artificial map hasarbitrary extendibility.) This will guarantee by Fact 3.2 that if the sequence Φi is derivedfrom from Φ0 := Φ by box inducing, then the moduli v(Φi) grow at least at a uniformlinear rate.

Lemma 4.2 also gives us an externally marked and uniformly quasiconformalbranchwise equivalence Υ0 from the phase space of φ to the phase space of Φ.Now proceed by complex pull-back as defined in the proof of Theorem 3, however dothe marking corrections by Proposition 4. We get a sequence of uniformly quasiconformalbranchwise equivalences between φi and Φ1.

Since quasiconformal mappings preserve com-plex moduli up to constants, v(φi) ≥Kv(Φi) with K only depending on the conformaldistortion of Υ0, thus ultimately (Lemma 4.2) only on the geometric bound of the holestructure of φ. Theorem 4 follows.50

4Construction of quasisymmetric conjugaciesHere is the main result of this section.Theorem 5 Suppose that f and ˆf are topologically conjugate and belong to some Fη. Assumealso that if f is renormalizable, then the first return time of its maximal restrictive intervalto itself is greater than 2.

Then for every δ > 0 there exist conjugate real box mappings,φe and ˆφe, either full or of type I and infinitely refined at the boundary, with a branchwiseequivalence υ between them so that the following list of properties holds:• φ and ˆφ are either suitable or δ-fine,• φ and ˆφ both have standard ǫ-extendibility,• υ is Q-quasisymmetric,• υ restricted to any long monotone domain replaces υ on the fundamental inducingdomain with distortion K,• υ restricted to any short monotone domain replaces υ on B with norm K.The numbers ǫ > 0, Q and K depend on η only.4.1Towards final mappingsTechnical details of the construction.The next lemma tells that given mappings φs,ˆφs and υs obtained from Theorem 2, we can modify an extend υs to an externally markedand fully internally marked quasiconformal branchwise equivalence.Lemma 4.1 Let φ and ˆφ be topologically conjugate complex box mappings, full or of type I,both infinitely refined at the boundary. Suppose that• both have hole structures geometrically bounded by K′,• on the boundary of each hole the mapping is K′′ quasisymmetric,• for the domain D of any branch of φ, |D|/dist(D, ∂J) ≤α holds with some fixed α;the same holds for every domain of ˆφ,• if the mappings are of type I, then |B′|/dist(B′, ∂J) ≤α and the same holds for ˆφ,• all long monotone branches of φ and ˆφ and ˆφ are ǫ-extendible, ǫ > 0,• υ exists which is a completely internally marked branchwise equivalence between φ andˆφ,• υ is Q-quasisymmetric,• υ restricted to any domain of φ replaces υ on J with distortion K.51

We claim that there a bound α0 depending on K′ only so that if α < α0, the followingholds:• φ and ˆφ can be extended to complex box mappings with diamonds, call them φd andˆφd,• an externally and completely internally marked branchwise equivalence Υ0 exists be-tween φd and ˆφd,• Υ0 is L-quasiconformal,• bounds L and α0 only depend on ǫ, K, K′, K′′ and Q.Proof:Let us first pick α0. Recall Fact 2.1 and choose κ as K1 picked for ǫ by this Fact.

Makeκ ≤1/2 as well. Consider the diamond neighborhood with height κ of J.

The bound α0should be picked so as to guarantee that all holes of φ, as well as the box B′ in case φ isof type I, sit inside this diamond neighborhood, moreover, that they have annular “collars”of definite modulus (say 1) which are also contained in this diamond neighborhood. Allholes quasidisks bounded in terms of K′.

So, there is a bounded ratio between how far theyextend in the imaginary direction and the length of the real domain they belong to. Nowit is evident that α0 small will imply this, and α0 depends only on K′.

Also, by making α0even smaller, we can ensure that diamond neighborhoods with height κ of all domains insideB′ are contained inside the complex box corresponding to B′, also with annular margins 1.Next, we choose the diamonds. We will take the diamond neighborhood with height κ ofJ as the complex box B0.

The diamonds will simply be preimages of this B0 by all monotonebranches. They will be contained in diamond neighborhoods with height κ of correspondingdomains of branches by the Poincar´e metric argument of [27].

Also, the diamonds will bepreimages of B0 with bounded distortion (Fact 2.1 again.) We can do the same thing for ˆφ.This gives us φd and ˆφd.The final step is to construct the branchwise equivalence by Lemma 1.2.

First, we decidethat on the real line the branchwise equivalence is υ. On the box around B0 extend it inany way that maps B0 onto ˆB0 and gives a uniformly quasisymmetric (in terms of Q and κ)mapping on the union of J and the upper (lower) half of B0.

Then Υ0 on the boundaries ofdiamonds is determined by pull-back. Observe, however, that on the curve consisting of theupper (lower) half of the diamond on the domain on the real line, the map is quasisymmetric,and that is because of the replacement condition.

It follows that diamonds can be filled withuniformly quasiconformal mappings as Lemma 1.2 demands.We first define Υ0 inside the complex box around B′. On the boundary of the box, wejust take a map transforming it onto the boundary of ˆB′ and quasiconformal with a boundednorm (in terms of K′ and Q) on the union of the upper (lower) half of the boundary of thecomplex box and the real box B′.

This propagates to the holes with bounded deteriorationof the quasisymmetric norm (in terms of K′′.) Thus, each hole can be filled with a uniformlyquasiconformal mapping.

Lemma 1.2 works to build Υ0 inside the complex box correspondingto B′ with desired uniformity. This having been achieved, Lemma 1.2 is again used insidethe entire complex box around B0.

Here, the complex box around B′ is formally regardedas a tooth. Finally, Υ0 is extended to the plane by quasiconformal reflection.52

Q.E.D.The initial branchwise equivalence.Suppose now that we are in the situation of The-orem 2 with mappings φs and ˆφs not suitable. We want to build an externally marked andcompletely internally marked branchwise equivalence Υs between them.

For that, we willuse Lemma 4.1 with φ := φs,b and ˆφ := ˆφs,b (the additional subscript b denotes versionsinfinitely refined at the boundary.) Comparing the assumptions of Lemma 4.1 with claims ofTheorem 2, we see that two conditions that are missing are the complete internal marking,and |D|/dist(D, ∂J) when D is a long monotone domain.

Let us first show that the secondproperty can be had by doing more inducing on long monotone domains. On the level ofinducing, we simply compose long monotone domains whose domains are too large with φs(ˆφs respectively) until we reduce their sizes sufficiently.

This can take many inducing stepson any given domain. Note that the hole structure can simply be pull-back.

The geometricbound of the hole structure will be worsened only in a bounded fashion provided that α wassmall enough. This follows from Fact 2.1.

Also, the replacement condition will not suffertoo much because we are pulling back by maps of bounded distortion (or one can formallyuse Proposition 1). The only hard point is the quasisymmetric norm.

This does not directlyfollow from Proposition 1, since we may have to do a large number of simultaneous monotonepull-backs. However, this is easily seen if we proceed by complex pull-back (like in the proofof Theorem 3).

To this end, we pick the diamond neighborhood of J with height 1/2 asB0, and a homothetic neighborhood of ˆJ as ˆB0. The diamonds are the preimages of B0 ( ˆB0resp.) by long monotone branches.

We do not have any holes. Then the argument used inthe proof of Lemma 4.1 applies and allows us to build a branchwise equivalence Υ′ “exter-nally marked” on the boundaries of all diamonds (but not on the boundaries of holes.) Wecan then perform the complex pull-back on long monotone branches any number of timeswithout increasing the quasiconformal norm.Next, we need to show that the complete internal marking can be realized with a boundedworsening of the bounds.

This follows directly from the proof of Lemma 2.1. This Lemmashows only how to implement the marking condition at the critical value, but the argumentworks the same way for any marking condition.The final maps.Now we can apply Lemma 4.1 to these modified mappings, and getφd, ˆφd and an externally marked completely internally marked branchwise equivalence Υdbetween them.

Now, they satisfy the hypotheses of Theorem 3. Proceed by complex inducingwith boundary refinement starting from φd and ˆφd.

It might be that full mappings occurinfinitely many times in this sequence. Otherwise, we can define final mappings φf, ˆφf withtheir branchwise equivalence Υf as either the initial triple φd, etc., if no full mapping occursin the sequence derived by complex inducing, or the last triple in this sequence with φf andˆφf full.

Observe that Theorem 3 is applicable with Υ = Υb = Υd. So, Υf has all propertiespostulated by Theorem 3 for Υ1.4.2Proof of Theorem 5Getting rid of boundary refinement.53

Lemma 4.2 Let φ be a box mapping, either full or of type I, infinitely refined at the bound-ary, which undergoes k steps of box inducing. Suppose that φ has standard ǫ-extendibility.Then there is a mapping φr obtained from φ by a finite number of simultaneous monotonepull-back steps using φ′ := φ so that after k box inducing steps starting from φr the resultingmap has standard ǫ-extendibility.Proof:The point is that box inducing skips boundary refinement.However, we show that thiscan be offset by doing enough “boundary refinement” before entering the box construction.Observe first the following thing.

Under the hypotheses of the Lemma, suppose that after kbox inducing steps we get a mapping φk and then do a simultaneous monotone pull-back onall long monotone branches of φk using φ′ := φk. Then the same mapping can be obtainedby doing a simultaneous monotone pull-back on all long monotone branches of the originalφ.

The proof of this remark proceeds by induction. For k = 1 this is rather obvious.

Forthe induction step from k −1 to k consider φ := φ1 and use the hypothesis of induction. Itfollows that we need to perform inducing on all long branches of φ1, and for that use the factagain with k = 1.

Now the lemma follows immediately, since each time one needs boundaryrefinement in a general inducing step, the appropriately refined mapping can be obtained bysimultaneous monotone pull-back on some or all long monotone branches of φ.Q.E.D.Main estimate.Lemma 4.3 Let φ and ˆφ be a pair of topologically conjugate complex box mapping with dia-monds of type I or full which undergo k steps of complex box inducing resulting in mappingsφk and ˆφk. Suppose that both hole structures can be assigned the separation index K. Sup-pose that box are infinitely refined at the boundary and have standard ǫ extendibility.

Also,suppose that a branchwise equivalence Υ exists which is Q-quasiconformal, externally markedand completely internally marked, and satisfies restricted to any long monotone domain onthe real line replaces Υ on the fundamental inducing domain with distortion K′. Then thereare numbers L1 which only depends on K and L2 depending on K′ and ǫ, with complexbox mappings Φ and ˆΦ of type I and a branchwise equivalence Υ′ between them so that thefollowing conditions are satisfied:• Υ′ is Q + L1-quasiconformal,• Υ′ is externally marked and completely internally marked,• Υ′ restricted to any long monotone domain replaces Υ′ on the fundamental inducingdomain with distortion L2,• Φ has the same box structure as φk, while ˆΦ has the same box structure as ˆφk,• Φ and ˆΦ both have standard ǫ-extendibility.54

Proof:The mapping Φ as obtained as φr for φ from Lemma 4.2. ˆΦ is obtained in the same way forˆφ.

They are topologically conjugate. By Lemma 4.2 this means that we should obtain someϕ0 by a series of simultaneous monotone pull-backs on long monotone branches of φ, andˆϕ0 is obtained in an analogous way for ˆφ.

Then we perform box inducing on ϕ0 , to get asequence ϕi with ϕk = Φ and the same is done for ˆϕ0 which gives ˆΦ = ˆϕk. The branchwiseequivalence is obtained by complex pull-back.Among the claims of Lemma 4.3 the extendibility is clear and the replacement conditionfollows in the usual way based on Lemma 4.6 of [18].

The hard thing is the quasiconformalestimate for Υ′. The procedure used in the proof of Theorem 3 does not give a uniformestimate for mappings which are not full.

However, by Proposition 4 and Theorem 4 modi-fications required to obtain the marking in the box case can be done with distortions whichdiminish exponentially fast at a uniform rate.Q.E.D.Conclusion.For the proof of Theorem 5, we begin by Theorem 2 which tells us that eitherwe hit a suitable map first, or we can build induced mappings φs, ˆφs and υs. If we encounterthe suitable map first, then the conditions of Theorem 5 follow directly from Theorem 2.Note that the assumption about the return time of the restrictive interval into itself is neededto make sure that the suitable mapping has monotone branches, and thus can be infinitelyrefined at the boundary.Otherwise, we proceed to obtain final maps with the branchwise equivalence betweenthem.

To this end, we build the complex branchwise equivalence, by Lemma 4.1, and pro-ceed by Theorem 3 to obtain the branchwise equivalence between final maps. If final mapsdo not exist, it means that infinitely many times in the course of the construction we ob-tain full mappings.

By Theorem 3, we get them with uniformly quasiconformal branchwiseequivalences. In this sequence of full mappings the sizes of domains other than long mono-tone ones go to 0.

So, having been given a δ we proceed far enough in the construction, andthen get the δ-fine mapping by applying simultaneous monotone pull-back on long monotonedomains. Theorem 5 follows in this case as well.So we are only left with the case when final maps exist.

Then we pick up the constructionby Lemma 4.3. Observe that the assumption about the separation index is satisfied for thefollowing reason.

The final map is either the same as φs, in which case the bound followsdirectly from Theorem 2, or is full and its holes are inside the holes of φs constructed byTheorem 2, so the separation index is even better. If f was renormalizable, we choose k inLemma 4.3 equal to the number of box inducing steps needed to get the suitable map.

Thenthe conditions of Theorem 5 follow directly from Lemma 4.3. The replacement conditionon short monotone domains is a consequence of the fact the by construction the branchwiseequivalence on short monotone domains is the pull-back of the branchwise equivalence fromB, and short monotone branches are extendible by Theorem 4.

When f is non-renormalizablewe choose a large k depending on δ and follow up with a simultaneous monotone pull-backon all long domains. Theorem 5 likewise follows.55

5Proof of Theorem 15.1The non-renormalizable caseTheorem 1 in the non renormalizable case follows directly from Theorem 5. Choose a se-quence δn tending to 0.

The corresponding branchwise equivalences obtained by Theorem 5for δ :deltan will tend to the topological conjugacy in the C0 norm. Since they are all uniformlyquasisymmetric in terms of η, so is the limit.5.2Construction of the saturated mapIn the renormalizable case, the only missing piece is the construction of saturated maps withquasisymmetric branchwise equivalences between them.

Also, we need to make sure that thebranches of the saturated map are uniformly extendible. The case when the return time ofthe maximal restrictive interval into itself is 2 is not covered by Theorem 5.

In this case, wesimply state that Theorem 1 is obvious and proceed under the assumption that the returntime is bigger than 2. So, Theorem 1 follows from this proposition:Proposition 5 Suppose that conjugate suitable real box mappings, ϕ and ˆϕ are given, botheither full or of type I and infinitely refined at the boundary, with a branchwise equivalenceΥ between them so that the following list of properties holds:• φ and ˆφ both have standard ǫ-extendibility,• Υ is Q-quasisymmetric,• Υ restricted to any long monotone domain replaces υ on the fundamental inducingdomain with distortion K,• Υ restricted to any short monotone domain replaces υ on B with norm K.Then, their saturated mappings ϕs and ˆϕs are ǫ-extendible.

Also a Q′-quasisymmetricsaturated branchwise equivalence Υ′ exists. Q′ depends on Q, K, and ǫ only.The proof of Proposition 5 is basically quoted from [18] with only minor adjustments.An outline of the construction.Let ψ mean the central branch.Let I denote therestrictive interval.

First, we want to pull the branches ϕ into the domain of ψ. We noticethat each point of the line which is outside of the restrictive interval will be mapped outsideof the domain of ψ under some number of iterates of ψ.

We can consider sets of points forwhich the number of iterates required to escape from the domain of ψ is fixed. Each such setclearly consists of two intervals symmetric with respect to the critical point.

The endpointsof these sets form two symmetric sequences accumulating at the endpoints of the restrictiveinterval, which will be called outer staircases. Consequently, the connected components ofthese sets will be called steps.56

This allows us to construct an induced map from the complement of the restrictive intervalin the domain of ψ to the outside of the domain ψ with branches defined on the steps of theouter staircases. That means, we can pull-back Υ to the inside of the domain of ψ.Next, we construct the inner staircases.

We notice that every point inside the restrictiveinterval but outside of the fundamental inducing domain inside it is mapped into the fun-damental inducing domain eventually. Again, we can consider the sets on which the timerequired to get to the fundamental inducing domain is fixed, and so we get the steps of apair of symmetric inner staircases.So far, we have obtained an induced map which besides branches inherited from ϕ hasuniformly extendible monotone branches mapping onto I.Denote it with ϕ1.We nowproceed by filling-in to get rid of short monotone branches.

We conclude with refinementof remaining long monotone branches. Thus, we will be left with branches mappings ontoJ only, so this is a saturated map.

Its extendibility follows from the standard argumentof inducing. The same inducing construction is used for ˆϕ.

Because the extendibility isobvious, we only need to worry the branchwise equivalence Υ′.Outer staircases.Suppose that the domain of ψ is very short compared with the lengthof the the domain of ϕ. This means that the domain of ψ is extremely large compared withthe restrictive interval.

This unbounded situation leads to certain difficulties and is dealtwith in our next lemma.Lemma 5.1 One can construct a branchwise equivalence Υ1 which is a pull-back of Υ withquasiconformal norm bounded as a uniform function of the norm of Υ. Furthermore, aninteger i can be chosen so that the following conditions are satisfied:• The functional equationΥ ◦ψj = ˆψjΥ1holds for any 0 ≤j ≤i whenever the left-hand side is defined.• The length of the interval which consists of points whose i consecutive images by ψremain in the domain of ψ forms a uniformly bounded ratio with the length of therestrictive interval.Proof:We rescale affinely so that the restrictive intervals become [−1, 1] in both maps.

Denotethe domains of ψ and ˆψ with P and ˆP respectively. Then, ψ can be represented as h(x2)where h′′/h′ is very small provided that |P| is large.

We can assume that |P| is large, sinceotherwise we can take Υ1 := Υ to satisfy the claim of our lemma. We consider the round diskB (“box”) whose diameter is the box of ϕ ranged through by ψ.

Because ψ is extendible,and its domain was assumed to be small compared to P, the preimage of B by φ, called B1sits inside B with a large annulus between them. Analogous objects are constructed for ˆϕ.It is easy to build a quasiconformal extension υ of Υ which satisfiesˆϕ ◦υ = υ ◦ϕon B1.

With that, we are able to perform complex pull-back by ψ and ˆψ.57

Also, assuming that |P| is large enough, we can find a uniform r so that the preimages ofB(0, r) by ψ, ˆ(ψ) and z →z2 are all inside B(0, r/2). Also, we can have B(0, r) contained inB1 as well as ˆB1.

Next, we choose the largest i so that [−r, r] ⊂ψ−i(P) . Then, we changeψ and ˆψ.

We will only describe what is done to ψ. Outside of B(0, r), ψ is left unchanged.Inside the preimage of B(0, r) by z →z2 it is z →z2.

In between, it can be interpolated bya smooth degree 2 cover with bounded distortion. The modified extension will be denotedwith ψ′.Next, we pull-back Υ by ψ′ and ˆψ′ exactly i times.

That is, if Υ0 is taken equal to Υ, thenΥj+1 is Υ refined by pulling-back Υj onto the domain of ψ. Now we need to check whetherΥ1 has all the properties claimed in the Lemma.

To see the functional equation condition,we note that all branches of any Υj are in the region where ψ coincides with ψ′. The lastcondition easily follows from the fact that r can be chosen in a uniform fashion.

Also, thequasiconformal norm of Υi grows only by a constant compared with Υ, since points passthrough the region of non-conformality only once.Q.E.D.The staircase construction.We take Υ1 obtained in Lemma 5.1 and confine our atten-tion to its restriction to the real line, denoted with υ1. We rely on the fact that υ1 is aquasisymmetric map and its qs norm is uniformly bounded in terms of the quasiconformalnorm of Υ1.Completion of outer staircases.We will construct a induced maps ϕ2 and ˆϕ2 witha branchwise equivalence υ2 with following properties:• The map υ2 coincides with υ1 outside of the domain of ψ.

Also, it satisfiesυ2 ◦ψj = ˆψj ◦υ2on the complement of the restrictive interval provided that ψj is defined.• Inside the restrictive interval, it is the “inner staircase equivalence”, that is, all end-points of the inner staircase steps are mapped onto the corresponding points.• Its qs norm is uniformly bounded as a function of the qc norm of Υ.Outer staircases constructed in Lemma5.1 connect the boundary points of the domain ofψ to the i-th steps which are in the close neighborhood of the restrictive interval. Also, thei-th steps are the corresponding fundamental domains for the inverses of ψ in the proximityof the boundary of the restrictive interval.By bounded geometry of renormalization, see [27], the derivative of ψ at the boundaryof the restrictive interval is uniformly bounded away from one.

Then, it is straightforwardto see that the equivariant correspondence between infinite outer staircases which uniquelyextends υ1 from the i-th steps is uniformly quasisymmetric (see a more detailed argumentin the last section of citekus.Inside the restrictive interval, the map is already determined on the endpoints of steps,and can be extended in an equivariant way onto each step of the inner staircase.58

Rebuilding a complex map.Due to the irregular behavior of the branchwise equiva-lence in the domains of branches of ϕ2 that map onto the central domain, they cannot befilled-in by critical pull-backs used in [18]. Instead, we will construct an externally markedbranchwise equivalence and apply complex filling-in.

The external marked will be achievedby Lemma 1.2. As the lip, we choose the circular arc which intersects the real line at theendpoints of the central domain and makes angles of π/4 with the line.

The teeth will be thepreimages of the lip by short branches inside the central domain. We check that the normof this mouth is bounded.

By the geodesic property (see [27]), the teeth are bounded bycorresponding circular arc of the same angle. The only property that needs to check is theexistence of a bounded modulus between the lip and any tooth.

This will follow if we provethat the intersection of a tooth with the real has a definite neighborhood (in terms of thecross-ratio) which is still inside the central domain. Since short branches inside the centraldomain are preimages of short monotone branches from the outside by a negative Schwarzianmap, it is enough to see the analogous property for the domains of short branches in thebox.

If the number of box steps leading to the suitable map was bounded, this follows fromthe estimates of [17], as the long monotone branches adjacent to the boundary of the boxcontinue to have a uniformly large size. Otherwise, one uses Theorem 3.Now we construct the branchwise equivalence on the lip.

To this end, we take the straightdown projection from the lip to the central domain, and lift the branchwise equivalence fromthe line. The resulting map on the lip is quasisymmetric with the norm bounded in termsof the quasisymmetric norm of ϕ2.

Now we pull-back this map to the teeth by dynamics.By the complex K¨obe lemma, the maps that we use to pull-back are diffeomorphisms ofbounded nonlinearity, so they will preserve quasisymmetric properties. Since we assumed thereplacement condition for short monotone domains, we can fill each tooth with a uniformlyquasiconformal map.

This leaves us in a position to apply Lemma 1.2 to fill the mouth witha uniformly quasiconformal branchwise equivalence. Call the mouth W.Finally, we have to extend the branchwise equivalence to the whole plane.

To this end,we choose a half-disk with the fundamental inducing domain normalized to [−1, 1] as thediameter, and make the branchwise equivalence identity there. Next, we regard W and allits preimages by short monotone branches as teeth.

This time, it is quite clear that the normis bounded. The branchwise equivalence of the teeth is pulled back from W by dynamics.The same argument as we made in the preceding paragraph show that Lemma 1.2 can beused to construct a complete branchwise equivalence on the plane.Construction of the saturated map.We now apply filling-in to all branches which maponto the central domain W. On the level of inducing, the only branches still left are thosewith the range equal to the fundamental inducing domain of the renormalized map, and longmonotone branches onto the whole previous fundamental inducing domain.Final refinement.We end with a simultaneous monotone pull-back on all long mono-tone branches.

The limit exists in L∞and is the saturated map. The branchwise equivalencewe get is quasiconformal as well.

This can be seen by choosing diamonds for all long mono-tone branches, marking it externally, and using complex pull-back. The paper [18] offers analternative way which does require external marking and instead relies on a version of the59

Sewing Lemma (Fact 2.2 in our paper.) This closes the proof of Proposition 5, and thereforeof all our remaining theorems.Appendix5.3Estimates for hole structuresSeparation symbols for complex box mappings.Definition of the symbols.Now, let ϕ be a type I complex box mapping of rank n.An ordered quadruple of real non-negative numbers:s(B) := (s1(B), · · ·, s4(B))will be said to give a separation symbol for B if certain annuli exist as described below.

Theannuli are either open or degenerate to curves. Figure 3 shows a choice of separating annulifor domain B, which is the same as domain B from Figure 2.We first assume that there are annuli A1(B) and A2(B).

Both annuli are contained in Bn′.The annulus A2(B) surrounds Bn separating it from the domain of the analytic extension ofB with range Bn′. Then A1(B) separates A2(B) from the boundary of Bn′.

We must haves2(B) ≤mod A2(B) ands1(B) ≤mod A2(B) + mod A1(B) .Next, three annuli are selected around B which will give the meaning of the two remainingcomponents of the symbol. First, the annulus A′(B) is chosen exactly equal to the differencebetween the domain of the canonical extension of the branch defined on B and the domainof B.

Then, the existence of A3(B) is postulated which surrounds A′(B) separating it fromBn and from the boundary of Bn′. Finally, A4(B) separates A3(B) from the boundary ofBn′.

Thens3(B) ≤mod A′(B) + mod A3(B) ands4(B) ≤mod A′(B) + mod A3(B) + mod A4(B) .The dependence on B will often be suppressed in our subsequent notations.Normalized symbols.We will now arbitrarily impose certain algebraic relations amongvarious components of a separation symbol. Choose a number β, and α := β/2, togetherwith λ1 and λ2.

Assume α ≥λ1, λ2 ≥−α and λ1 + λ2 ≥0. If these quantities are connectedwith a separation symbol s(B) as followss1(B) = α + λ1 ,60

s2(B) = α −λ2 ,s3(B) = β −λ1 ,s4(B) = β + λ2 .we will say that s(B) is normalized with norm β and corrections λ1 and λ2.Separation index of a box mapping.For a type I complex box mapping φ a positivenumber β is called its separation index provided that valid normalized separation symbolswith norm β exist for all univalent branches.Monotonicity of separation indexes.The nice property of separation indexes is thatthey do not decrease in the box inducing process. In fact, one could show that they increaseat a uniform rate and this will be the final conclusion to be drawn from Theorem 4.

Fornow, we proveLemma 5.2 Let φi, i between 0 and m be a sequence of complex box mappings of type I withthe property that the next one arises from the previous one in a simple box inducing step. Ifβ0 is a separation index of φ0, then β0 is also a separation index of φi for any i < m. Inaddition, if φi arises after a non-close return, then v(φi) ≥β0/4.Proof:The proof of Lemma has to be split into a number of cases.

As analytic tools, we will use thebehavior of moduli of annuli under complex analytic mappings. Univalent maps transportthe annuli without a change of modulus, analytic branched covers of degree 2 will at worsthalve them, and for a sequence of nesting annuli their moduli are super-additive (see [19],Ch.

I, for proofs, or [5] for an application to complex dynamics.) To facilitate the discussion,we will also need a classification of branches depending on how they arise in a simple boxinducing step.Some terminology.Consider a abstract setting in which one has a bunch of univalentbranches with common range B′ and fills them in to get branches mapping onto some B ⊂B′.The original branches mapping onto B′ will be called parent branches of the filling-in process.Clearly, every branch after the filling-in has a dynamical extension with range B′.

For twobranches, the domains of these respective extensions may be disjoint or contain one another.In the first case we say that the original branches were independent. Otherwise, the onemapped with a smaller extension domain is called subordinate to the other one.

Note that ifb′ is subordinate to b, then the extension of b maps b′ onto another short univalent domain.We then distinguish the set of ”maximal” branches subordinate to none.They aremapped by their parent branches directly onto the central domain. Therefore, the domainsof extensions of maximal branches mapping onto B′ are disjoint.

They also cover domains ofall branches. The extensions of maximal branches are exactly parent branches of the filling-inprocess.

These extensions with range B′ will called canonical extensions.Now, in a simple box inducing step, the parent branches are the short monotone branchesof ˜φ. Among these we distinguish at most two immediate branches which restrictions of the61

central branch of φ to the preimage of the central domain. All non-immediate parent branchesare compositions of the central branch of φ with short monotone branches of φ.

For example,in the non-close return the first filling gives a set of parent branches, two of which may beimmediate, which later get filled in. In the close return filling-in is done twice, so we will bemore careful in speaking about parent branches.

Figure 2 shows examples of independentand subordinate domains.We will sometimes talk of branches meaning their domains, for example saying that abranch is contained in its parent branch. We assume that φ has rank n, so B = Bn andB′ = Bn′.

Let ψ be the central branch of φ. Let Bn+1 denote the central domain of thenewly created map φ1.

Observe that B(n+1)′ = Bn. Suppose that β is a separation indexof φ.

We will now proceed to build separation symbols with norm β for all short univalentdomains of φ1. Let g be a short univalent branch of φ1 and p denote the parent branch g.The parent branch necessarily has the form P ′ ◦ψ.

Let P be the branch of φ whose domaincontains the critical value. Objects (separation annuli, components of separation symbols)referring to φ1 will be marked with bars.Reduction to maximal branches.Note that it is sufficient show that symbols withnorm β exist for maximal branches.

Indeed, suppose that a separation symbol exists for amaximal branch b and let b′ be subordinate to b. We can take A1(b′) = A1(b) and A2(b′) =A2(b).

Likewise, we can certainly adopt A4(b′) = A4(b), and A3(b′) can be chosen to containA3(b). The annulus A′(b) is the preimage of the annulus Bn′ \ Bn by the parent branch of b.The annulus A′(b′) is the preimage of the same annulus by the canonical extension of b, so ithas the same modulus.

Since the domain of the canonical extension of b′ is contained in theparent domain (equal to the domain of the canonical extension of b), the assertion follows.Non-close returns.Let us assume that φ makes a non-close return, that is P ′ ̸= ψ.The case of p immediate.Let b denote the maximal branch in p. The new central holeBn+1 is separated from the boundary of Bn by an annulus of modulus at least (β +λ2(B))/2.The annulus A2(b) around Bn+1 will be the preimage by the central branch of the regioncontained in and between A3(P) and A′(P). Then, A1(b) is the preimage of A4.

It followsthat we can takes1 = β + λ2(P)2ands2 = β −λ1(P)2.Of course, since components of the symbol are only lower estimates, we are always allowedto decrease them if needed. The annulus A′ is naturally given as the preimage of the annulusbetween Bn+1 and the boundary of Bn by the central branch, likewise A3 is the preimageof A2(P), and A4 is the preimage of A1(P).

Since the first two preimages are taken in anunivalent fashion, we gets3 = β + λ2(B)2+ α −λ2(B) and62

s4 = s3 + λ1(B) + λ2(B)2= β2 + α + λ1(B)2.Thus, if we putλ1 = λ2(B)2, λ2 = λ1(B)2we get a valid separation symbol with norm β. In the remaining non-immediate cases, thebranch P ′ is defined.P ′ and P non-immediate and independent.To pick A2(b), we take the preimageby ψ of the annulus separating P from the boundary of the domain of its canonical extensionwith range Bn′, i.e.

A′(P). We claim that its modulus in all cases is estimated from below byα+δ where δ is chosen as the supremum of −λ2(b′) over all univalent domains b′ of φ. Indeed,P is carried onto Bn by the extended branch, and the estimate is α plus the maximum ofλ1(b′) with b′ ranging over the set of all short univalent domains of φ. and λ1(P ′) which isat least The assertion follows since λ1(b′) + λ2(b′) ≥0 for any b′.

To pick A1(b), consider theannulus separating A′(P) from the boundary of Bn′, i.e. the region in and between A3(P)and A4(P).

Pull this region back by the central branch to get A1(b). By the hypothesis ofthe induction, the estimates ares1 = β + λ2(P)2ands2 = α + δ2.Since b is maximal A′(b) is determined with modulus at least s1.

The annulus A3(b) willbe obtained as the preimage by the central branch of A′(P ′). This has modulus at least α+δin all cases as argued above.

The annulus A4(b) is the preimage of the region in and betweenA3(P ′) and A4(P ′). By induction,s3 = β + λ2(P)2+ α + δ ands4 = s3 + β + λ2(P ′) −α −δ2.We put λ1 = λ2(P )2and λ2 = α−δ2 .

We check thats3 + λ1 = β2 + α + λ2(P) + δ ≥β −λ2(P) + λ2(P) ≥β .In a similar way one verifies thats4 −λ2 ≥β .Also, the required inequalities between corrections λi follow directly.63

P ′ subordinate to P.This means that some univalent mapping onto Bn′ transformsP onto Bn and P ′ onto some P ′′. Consider A2(P ′′) which separates Bn from P ′′, and a largerannulus A1(P ′′).Their preimages first by the extended branch and then by the centralbranch give us A2(b) and A1(b) respectively.

The estimates ares2 = α −λ2(P ′′)2ands1 = α + λ1(P ′′)2.The annulus A′(b) is uniquely determined with modulus s1, and A3(b) will be the preimageof the annulus separating P ′′ from Bn. Finally, A4(b) will separate the image of A3(b) fromBn′.

The estimates ares3 = α + λ1(P ′′)2+ β −λ1(P ′′) = β + α −λ1(P ′′)2ands4 = s3 + λ1(P ′′) + λ2(P ′′)2= β + α + λ2(P ′′)2.Setλ1 = −α + λ1(P ′′)2andλ2 = α + λ2(P ′′)2.The requirements of a normalized symbol are clearly satisfied.P subordinate to P ′.This situation is analogous to the situation of immediate parentbranch considered at the beginning. Indeed, by mapping P ′ to Bn and composing with thecentral branch one can get a folding branch with range Bn′ defined on P ′.

We now see thatthe situation inside the domain of the canonical extension of P is analogous to the case ofimmediate parent branches, except that the folding branch maps onto a larger set Bn′. Sothe estimates can only improve.A close return.In this case there are no immediate parent branches and we really haveonly one case to consider.

Fix some short univalent branch b of φ1, let p be its parent branch,and denote p = P ′ ◦ψ. Consider A2(P ′) and A1(P ′).

Their preimages by the central branchgive us A2(b) and A1(b) respectively. The estimates ares2 = α −λ2(P ′)2ands1 = α + λ1(P ′)2.The annulus A′(b) is uniquely determined with modulus s1, and A3(b) will be the preimageof the annulus separating P ′ from Bn i.e.

the annulus containing A3(P ′) and A′(P ′) together64

with the region between them. Finally, A4(b) will be the preimage of A4(P ′) by ψ. Theestimates ares3 = α + λ1(P ′)2+ β −λ1(P ′) = β + α −λ1(P ′)2ands4 = s3 + λ1(P ′) + λ2(P ′)2= β + α + λ2(P ′)2.Setλ1 = −α + λ1(P ′)2andλ2 = α + λ2(P ′)2.The requirements of a normalized symbol are clearly satisfied.

Not quite surprisingly,these are the same estimates we got in the non-close case with P ′ subordinate to P.Conclusion.We already proved by induction that β0 remains a separation index for allφi. It remains to obtained the estimate v(φi+1) ≥β0/4 under the assumption that φi makesa non-close return.

This is quite obvious from considering the separation symbol for thebranch P which contains the critical value. Since s4(P) = β0 + λ2(P) geqβ0/2 and becauseof superadditivity of conformal moduli, there is an annulus with modulus at least β0/2separating P from B′, and its pull-back by ψ gives as an annulus with desired modulus.

Thisall we need to finish the proof of Lemma 5.2. Note, however, that we cannot automaticallyclaim s1 ≥β0/4 even if the preceding return was non-close.Q.E.D.References[1] Ahlfors, L.V.

: Quasiconformal reflections, Acta Math. 109 (1963), 291-301[2] Ahlfors, L.V.

& Sario, L.:Riemann surfaces, Princeton University Press, Princeton NJ(1960)[3] Beurling, A. & Ahlfors, L.: The boundary correspondence under quasiconformal map-pings, Acta Math.

96 (1956), pp. 125-142[4] Blokh, A.

& Lyubich, M.: Non-existence of wandering intervals and structure of topo-logical attractors for one dimensional dynamical systems, Erg. Th.

& Dyn. Sys.

9 (1989),751-758[5] Branner, B. & Hubbard, J.H.

: The iteration of cubic polynomials, Part II: patterns andparapatterns, to appear in Acta Math. [6] Douady, A.

& Hubbard, J.H. : On the dynamics of polynomial-like mappings, Ann.

Sci.Ec. Norm.

Sup. (Paris) 18 (1985), pp.

287-343[7] Fatou, P.:Sur les equations fonctionnelles, II, Bull. Soc.

Math. France, 48 (1920), 33-9465

[8] Gardiner, F.: Teichm¨uller theory and quadratic differentials, John Wiley & Sons, NewYork (1987)[9] Graczyk, J. & Swiatek, G.:Critical circle maps near bifurcation, Stony Brook IMSpreprint (1991)[10] Graczyk, J.

$ ´Swi¸atek, G.: Induced expansion for quadratic polynomials, preprint[11] Guckenheimer, J.: Limit sets of S-unimodal maps with zero entropy, Commun. Math.Phys.

110 (1987), pp. 655-659[12] Guckenheimer, J.: Sensitive dependence on initial conditions for one dimensional maps,Commun.

Math. Phys.

70 (1979), pp. 133-160[13] Guckenheimer, J.

& Johnson, S.: Distortion of S-unimodal maps, Annals of Math. 132,71-130 (1990)[14] Hubbard, J.H.

: Puzzles and quadratic tableaux, manuscript[15] Jakobson, M.: : Absolutely continuous invariant measures for one-parameter familiesof one-dimensional maps, Commun. Math.

Phys. 81 (1981), pp.

39-88[16] Jakobson, M.: Quasisymmetric conjugacies for some one-dimensional maps inducingexpansion, Contemporary Mathematics 135, 203-211 (1992)[17] Jakobson, M. & ´Swi¸atek, G.: Metric properties of non-renormalizable S-unimodal maps,preprint IHES no. IHES/M/91/16 (1991)[18] Jakobson, M. & ´Swi¸atek, G.: Quasisymmetric conjugacies between unimodal maps,Stony Brook preprint 16 (1991)[19] Lehto,O.&Virtanen,K.

:QuasikonformeAbbildungen,Springer-Verlag,Berlin-Heidelberg-New York (1965)[20] Lyubich, M,, Milnor, J.:The dynamics of the Fibonacci polynomial, Stony Brookpreprint (1991)[21] Ma˜ne, R. , Sad, P. & Sullivan, D.: On the dynamics of rational maps, Ann. Ec.

Norm.Sup. 16 (1983), pp.

193-217[22] McMullen, C.: Complex dynamics and renormalization, preprint, Berkeley (1993)[23] de Melo W., van Strien S.One-Dimensional Dynamics, Springer-Verlag, New York(1993)[24] Milnor, J.: The Yoccoz theorem on local connectivity of Julia sets. A proof with pic-tures., class notes, Stony Brook, (1991-92)[25] Milnor, J.

& Thurston, W.: On iterated maps of the interval I, II, preprint (1977), in:Lecture Notes in Mathematics, Vol. 1342 (Springer, Berlin, 1988), p. 46566

[26] Preston, C.: Iterates of maps on an interval, Lecture Notes in Mathematics, Vol. 999.Berlin,Heidelberg,New York: Springer (1983)[27] Sullivan, D.: Bounds, quadratic differentials and renormalization conjectures, in Math-ematics in the twenty-first century, American Mathematical Society, Providence, RI(1991)[28] Yoccoz, J.-C.: unpublished results, see the discussion in [14] and [24]67

B0BnBn’BnFigure 1: A sample graph of a type I real box mapping. All three permissible types ofbranches: central folding, long and short monotone are shown.

Be aware that typically onehas infinitely many branches.

B D1D2DD0B’Figure 2: A type I complex box mapping. Dotted lines show domains of canonical extensions.Domains D0 and D are look like they are maximal.

Then D1 and D2 are subordinate toD0, but apparently independent from one another as well as from D. D0 and D are alsoindependent. There may be univalent domains outside of B′, not shown here.

B nB nB n’B nBA12ABA 43A’A’Figure 3: A choice of separating annuli for B. Note that the outermost annuli A1 and A4are filled in white.

---

출처: arXiv:9207.219 • 원문 보기