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Combinatorics, geometry and attractors of quasi-quadratic maps.

arXiv:math/9212210v1 [math.DS] 6 Dec 1992Combinatorics, geometry and attractors of quasi-quadratic maps.Mikhail Lyubich*Mathematics Department and IMS, SUNY Stony BrookFebruary 1992, revised October 1992Abstract. The Milnor problem on one-dimensional attractors is solved for S -unimodal maps with a non-degenerate critical point c .

It provides us with a complete understanding of the possible limit behaviorfor Lebesgue almost every point. This theorem follows from a geometric study of the critical set ω(c) of a“non-renormalizable” map.

It is proven that the scaling factors characterizing the geometry of this set godown to 0 at least exponentially. This resolves the problem of the non-linearity control in small scales.

Theproofs strongly involve ideas from renormalization theory and holomorphic dynamics.§1. Introduction.Let f : [0, 1] →[0, 1] be an S -unimodal map (see the definitions later) of the interval with a non-degenerate critical point c .

Let us call such a map quasi-quadratic. As usual, ω(x) denotes the limit set ofthe forward orb (x) .

The following theorem solves the Milnor problem [M1].Theorem on the Measure-Theoretic Attractor. Let f be a quasi-quadratic map normalized by thecondition f : c 7→1 7→0 .

Then there is a unique set A (a measure-theoretic attractor in the sense of Milnor)such that A = ω(x) for Lebesgue almost all x ∈[0, 1] , and only one of the following three possibilities canoccur:1 . A is a limit cycle;2 .

A is a cycle of intervals;3 . A is a Feigenbaum-like attractor.This result gives a clear picture of measurable dynamics for the maps under consideration.

Let usexplain the words used in the statement. A limit cycle is the periodic orbit whose basin of attraction hasnon-empty interior.

A cycle of intervals is the union of finitely many intervals In with disjoint interiorscyclically interchanged by the dynamics.A Feigenbaum-like attractor is an invariant Cantor set of thefollowing structure:A =\On,where O1 ⊃O2 ⊃... is a nested sequence of cycles of intervals of increasing periods.It makes sense to compare the above Theorem with its topological counterpart known since late 70s:Theorem on the Topological Attractor ([MT], [G], [JR], [vS]). Let f be an S -unimodal mapnormalized by the condition f : c 7→1 7→0 .

Then there is a unique set Λ (a topological attractor) suchthat Λ = ω(x) for a generic x ∈[0, 1] , and only one of the following three possibilities can occur:( i). Λ is a limit cycle;( ii).

Λ is a cycle of intervals;( iii). Λ is a Feigenbaum-like attractor.From this point of view the Theorem on the Measure-Theoretic Attractor says that the map f has aunique measure-theoretic attractor A coinciding with the topological attractor Λ .

In cases (i) and (iii) thiswas proven by Guckenheimer [G]. In case (ii) it is known thatΛ =p−1[k=0Ikwhere Ii form a cycle of intervals of period p , and f ◦p| Ik is topologically exact.

The last property meansthat for any interval J ⊂Ik there is an n such that f ◦pnJ = Ik . So, we can reduce the Theorem on theMeasure-Theoretic Attractor to the following statement:* Supported in part by NSF grant DMS-8920768 and a Sloan Research Fellowship.1

Theorem A. Let f be a quasi-quadratic map.

Assume that f is topologically exact on [0, 1] , henceΛ = [0, 1] . Then ω(x) = [0, 1] for Lebesgue almost all x ∈[0, 1] , that is A = [0, 1] .This measure-theoretic result follows from geometric properties of the critical set ω(c) .

In order tostudy microstructure of this set we introduce the concept of a generalized renormalization as the rescaledfirst return map restricted to a neighborhood of the critical set. This moves us out of the class of unimodalmaps to a class T of maps with a single critical point but defined on the union of disjoint intervals mappedonto a bigger interval.

†Any map with recurrent critical point becomes infinitely renormalizable in thissense.Let us consider the sequence of renormalized mapsR◦nf :[iIni →In−10.As basic geometric characteristics of the critical set we consider the scaling factors µn = |In0 |/|In−10| andthe Poincar´e lengths of the gaps between the intervals Iniof level n . Our main geometric result isTheorem B.

Let f be a topologically exact quasi-quadratic map with the recurrent critical point. Thenthe Poincar´e lengths of the gaps of the renormalized maps go up to ∞.Moreover, to a first approximation they increase at least linearly.

Unfortunately, there is one unpleasantcircumstance which can slow the rate down, namely the long cascades of “central returns” (when the map iscombinatorially close to being renormalizable). However, the rate is still under explicit control.

In particular,if we number these cascades by κ ∈N then the rate will be at least linear in κ .As it is clear from the very name of our field, the basic problem of non-linear dynamics is to gain controlof non-linearity. In our situation the non-linearity in small scales is controlled by the above mentioned scalingfactors.

Theorem B implies that the scaling factors go down at least exponentially in κ . This provides uswith a perfect control of non-linearity: the high order renormalized maps are becoming purely quadraticexponentially fast.

Observe that this contrasts drastically with the Feigenbaum-like case when the geometryof the critical set is bounded from below (provided the combinatorics is bounded) which creates a definiteamount of non-linearity in all scales (see Sullivan [S]).Let us now dwell in more detail on the ideas of this work. §2 contains the combinatorial treatment ofmaps of class T .

According to the recurrent properties of the orb( c ) we split the analysis into two subcases:the reluctantly and the persistently recurrent. The latter case presents a stronger recurrence of the criticalpoint.

For example, in this case the return time of points of the orb (c) back to a neighborhood U ∋c isbounded. Surprisingly this helps to provide the further analysis since the renormalized maps turn out to beof finite type (that is, defined on the finitely many intervals).

We show that such maps are classified by thecascades of renormalization types, and that these types can be combined independently. The last statementgives us a big freedom in producing examples.

In particular, the so called Fibonacci map naturally arisesas a map of the simplest stationary type. This map studied in [LM] was a basic model which clarified thesituation.§§3-6 are occupied with the proof of Theorem B.

In §3 we prove it under the assumption that we startwith a small enough scaling factor. To this end we introduce a modification of the Poincar´e lengths, the“asymmetric Poincar´e lengths”, which behave more regularly under renormalization.

We show that theasymmetric Poincar´e lengths of the gaps increase almost monotonically. Though this is the longest technicalpiece of the paper, we are actually in a quite comfortable position since the starting condition provides usat once with a perfect control of non-linearity.In order to get rid of the starting assumption we pass to the complex plane in the class of (generalized)polynomial-like maps, a complex counterpart of maps of class T(§6).These maps are defined on thefinite union of disjoint topological disks mapped onto a bigger disk.

The crucial fact is that all such mapswith the same combinatorics are quasi-conformally equivalent which yields quasi-symmetric equivalence onthe real line. This part relies on recent developments in holomorphic dynamics, particularly on the work ofBranner and Hubbard [BH].

Since the conclusion of Theorem B is quasi-symmetrically invariant, it is enough† Up to some point we allow the domain to consist of infinitely many intervals. However, the renormal-ization philosophy becomes really valuable only in the finite case.2

to produce just one example of a polynomial-like map f with a given combinatorics and arbitrarily smallstarting scaling factor. Exactly at this point it is important to have a freedom gained by passing to the classof generalized polynomial-like maps and its real counterpart, class T .A bridge between quadratic maps and generalized polynomial-like maps is given by the renormalizationon a Yoccoz puzzle-piece introduced in [L2], a complex analogue of the renormalization mentioned above.This passage completes the proof of Theorem B for quadratic polynomials (see the end of §6).

The argumentfor general quasi-quadratic maps is based upon two Sullivan’s Principles [S]:1 ) Renormalizations of sufficiently smooth maps are becoming real analytic;2 ) High order renormalization of a real analytic map makes it polynomial-like.These principles turn out to be efficient in our setting of generalized renormalization. The First Principleworks when we have combinatorics of bounded type, that is the number of intervals Iniis bounded on alllevels.

To push it forward we need a priori distortion bounds obtained by Martens [Ma]. As to the case ofunbounded combinatorics (as well as the non-minimal case), it is actually easier, and can be treated by apurely real argument.

We discuss these issues in §4.In §5 we push the Second Principle forward. In order to construct a polynomial-like map we considerthe complex pull-backs of Euclidian disks based upon the real intervals, and estimate their sizes (“complexbounds”).

This is another technical piece of the work.Finally, in the last §7 we derive Theorem A from Theorem B by showing that the set X = {x : ω(x) ∋c}has zero lower density at c . This is our first application of geometric Theorem B but we expect a lot ofothers.Remarks.1.

Since Milnor’s paper of 1985 there has been a good deal of effort to attack the problemof attractors, see [BL1-3], [GJ], [K], [Ma], [JS]. In particular, it was already known that there is only onemeasure-theoretic attractor (see [BL1-3] or [GJ], [K]).

The reluctantly recurrent case was resolved in [BL3]and [GJ].The geometric study of “non-renormalizable maps” was started in [GJ], [Ma] and [JS]. In [JS] the absenceof Cantor attractors was proved for topologically exact maps sufficiently close to the Chebyshev polynomialx 7→4x(1 −x) .In [HK] the Fibonacci map was suggested as a candidate for a “wild” situation when the measure-theoretic and topological attractors are different.

The paper [LM] showed that it is not the case for quasi-quadratic maps (an alternative purely real argument has been recently found in [KN]). However, a computerexperiment carried out jointly with F.Tangerman indicates that this may well be true in higher degrees.For a survey on the problem of attractors see [L3].2.

The complex counterpart of Theorem A says that the Lebesgue measure of the Julia set of a “non-tunable” quadratic polynomial is equal to 0 (Lyubich [L2] and Shishikura (unpublished)). The case of cubicpolynomials with Cantor Julia set had been earlier treated by McMullen (see [BH]).

The real result turns outto be harder than the complex one because conformal invariants (like annulus moduli or Poincar´e metric)have only quasi-invariant analogues in the real setting. However, Theorem B yields that on deep levels theyare becoming invariant exponentially fast with respect to pull-backs.

This makes the real-complex dictionarymuch more precise.3. There are two pieces of the paper which strongly depend on the quadratic-like nature of the criticalpoint.

First, the growth of the Poincar´e lengths of the gaps relies on the fact that the square root mapdivides the Poincar´e length at most by 2 (Lemma 3.8). (By the way, this is the place where the asymmetricPoincar´e length comes to the scene).

The second place is the Branner-Hubbard divergence property (§6).Some definitions, notations and conventions. A continuous map f : [0, 1] →[0, 1] is called unimodalif it has only one extremum c .

It is called S -unimodal if additionally it is three times differentiable withonly critical point c and with negative Schwarzian3

derivative outside c :Sf = f ′′′f ′ −32f ′′f ′2< 0.If additionally the critical point c is non-degenerate, that is f ′′(c) ̸= 0 , then the map is called quasi-quadratic.Let φ(x) = (x −c)2 . We use the abbreviations qs and qc for “quasi-symmetric” and “quasi-conformal”respectively.Saying “an interval” we mean a “closed interval”.

The notation I = [a, b] means that a and b are theendpoints of I but does not necessarily mean that a < b . Points a and a′ are called “ c -symmetric” iffa = fa′ .

So, we can also talk about c -symmetric sets.The n -fold iterate of a map g is denoted by g◦n . The orb( x) ≡g - orb(x) denotes {g◦mx}∞m=0 .

Alsoorb n(x) = {g◦mx}nm=0 denotes the initial piece of the orbit. Let N = {0, 1, 2, ...} .We recommend the forthcoming book of de Melo and van Strien [MS] for the background in one-dimensional dynamics.Acknowledgement.

I’d like to thank Marco Martens for useful discussions, and for reading parts of themanuscript. I owe to John Milnor several nice suggestions which improved the exposition.

I also take thisopportunity to thank IHES and IMPA for their hospitality while I was doing parts of this work.§2. Renormalization.Renormalization in dynamical systems means the first return map to an appropriate piece of the phasespace and then rescaling of this piece to the “original size”.

For example, if we have a periodic c -symmetricinterval I ⊂[0, 1] of period p > 1 , we can consider the unimodal map f p|I and then rescale I backto [0,1]. This is the usual notion of renormalization in one-dimensional dynamics.

So, we can talk aboutrenormalizable and non-renormalizable maps. Repeating this procedure we can further talk about “twicerenormalizable”, “thrice renormalizable”,..., or at the end “infinitely renormalizable” maps.

For example,looking through the Theorem on the Topological Attractor we see that case (ii) corresponds to at mostfinitely renormalizable maps, while the last case (iii) corresponds to infinitely renormalizable maps. Thecase we concentrated on, when the map f is topologically exact on [fc, f 2c] , is equivalent to being non-renormalizable.However, this specific terminology highly accepted in one-dimensional dynamics is somewhat misleading.Indeed, we will see that the most interesting “non-renormalizable” maps can actually be treated as infinitelyrenormalizable in an appropriate sense, and this gives an efficient tool for studying the geometry of the criticalset.

This renormalization does not respect the class of unimodal maps. So, let us describe an appropriateclass of maps.Class T .

Let q, p ∈[0, ∞] , and let Ik, k = −q, ..., p, be a finite family of disjoint intervals compactlycontained in a base interval J . The interval I0 is marked, and will be called central.

Let us consider a mapg : Dom(g) =p[k=−qIk →J(2 −1)with a single turning point c ∈I0 (we will still call it “critical”), which also satisfies the following property:g(∂Ik) ⊂∂J . In particular, g maps each non-central interval Ik, k ̸= 0 , homeomorphically onto the wholeinterval J , while the central interval I0 is c -symmetric.Let us also assume that g does not have non-trivial wandering intervals.

This condition holds automat-ically under some regularity assumptions, e.g., g has negative Schwarzian derivative, and the critical pointc is non-flat (see [G] or [L1]).Denote this class of maps by T ∞. Let us say that g ∈T ∞is of finite type if its domain consists ofonly finitely many intervals, that is p, q ∈N .

Let T denote the subclass of maps of finite type. Unimodalmaps can also be viewed as maps of class T whose domain contain a single interval.A point x is said to be non-escaping if g◦nx ∈Dom(g) for all n = 1, 2, ... .

Denote by K(g) the setof non-escaping points ( the “filled-in Julia set” of g ).4

Pull-Backs and nice intervals.The following pull-back construction plays an essential role in whatfollows. Given an interval T and a point x such that g◦nx ∈intT , we can pull T back along the orbit{g◦kx}nk=0 .

This means that we inductively construct a sequence of intervalsTn ≡T ∋g◦nx,Tn−1 ∋g◦(n−1)x,..., T0 ∋xso that Tk is the maximal interval containing gkx whose image is contained in Tk+1 . An interval Tk ofthe pull-back is called critical if int Tk ∋c .

Since g(∂Ik) ⊂∂J ,g(∂Tk) ⊂∂Tk+1, k = 0, 1, ..., n −1. (2 −2)Hence the non-critical intervals Tk diffeomorphically map onto Tk+1 , and the critical intervals Tk arec -symmetric, k = 0, 1, ..., n −1 .

A pull-back is called monotone if all intervals Tk except perhaps T arenon-critical. It is called unimodal if T0 is critical while T1, ..., Tn−1 are non-critical.As in [Ma] let us call a c -symmetric interval T nice ifg◦n(∂T ) ∩T = ∅, n = 1, 2, ...(2 −3)For example, let α be a periodic point, and let β be its n -fold preimage, β′ be a c -symmetric point.Then the interval T = [β, β′] is nice provided β ̸∈orb(α) .

It follows that there are nice intervals in anyneighborhood of c (provided g has no limit cycles).Let us denote by M ≡MT the family of intervals obtained by pulling a nice interval T back along allpossible orbits ( M comes from “Markov”). Following the analogy with the holomorphic setting (compare[H], [M2] or §6), the intervals of M will also be called puzzle-pieces.

We use the notation T (n)(x) forthe puzzle-piece obtained by pulling T along orb n(x) , and call n the depth of the puzzle-piece. (If westart with another interval, say, J then, of course, we use the notations J(n)(x) for the correspondingpuzzle-pieces).

The basic properties of the family M are:(i) any two intervals of M are either disjoint or “strongly nested”; in the latter case the interval of higherdepth is contained in the interior of the other one;(ii) any non-critical puzzle-piece T (n)(x) diffeomorphically maps onto T (n−1)(fx) . (iii) For any critical puzzle-piece T (n)(x) of depth > 0f(T (n)(x), ∂T (n)(x)) ⊂(T (n−1)(fx), T (n−1)(fx)).All critical puzzle-pieces are c -symmetric.

(iv) If c is recurrent then for any puzzle-piece T (n)(x) , the orb (c) does not cross ∂T (n)(x) .Lemma 2.1. Let K ∈M be a critical puzzle-piece, and x be a point whose orbit crosses int K .

Take thefirst moment n > 0 for which g◦nx ∈intK . Then the pull-back of K along orb( x ) is either monotone orunimodal.Proof.

Let K = Km, Km−1, ..., K0 ∋x be the above pull-back. Then the puzzle-pieces Km−1 ,..., K0 havehigher depths than K .

It follows that they are are disjoint from K and hence non-critical. ⊔⊓In particular, if x ̸∈K then the pull-back is monotone.

If x = c is the critical point then the pull-backis certainly unimodal.Let K, L ∈M be two critical intervals, L ⊂K . The interval L is called a kid of K if it is obtainedby a unimodal pull-back along a piece of the critical orbit {g◦kc}nk=0, gnc ∈K .

The kid corresponding tothe first return of the critical point back to K will be called the first kid K1 . Lemma 2.1 can be improvedin the following way.Lemma 2.2.

Let K ∈M be a critical puzzle-piece, L be its first kid, and x be a point whose orbitcrosses int L . Take the first moment n > 0 for which g◦nx ∈intL .

Then the pull-back of K along orb( x )is monotone if x ̸∈L , and unimodal otherwise.Proof. Consider the subsequent return moments 0 < n1 < n2 < ... < nl = n of orb (x) to K until thefirst meeting with L .

According to the previous lemma, the first return produces a unimodal or monotonepull-back of K depending on whether x ∈L or not. All further returns produce monotone pull-backs ofK .

This is what was required. ⊔⊓5

Sequence of first grandkids. Cascades of central returns.

Let I0 ≡J be the base interval, I1 ≡I0be its first kid, I2 be the first kid of I1 etc. In such a way we construct a nested sequence of the “firstgrandkids”I0 ⊃I1 ⊃I2 ⊃...which will play fundamental role in what follows.

Let t(n) be the first return time of the critical point backto In−1 . We say that the return to level n −1 is central if g◦t(n)c ∈In .

Otherwise the return is classifiedas high or low depending on whether g◦t(n)In ⊃In or g◦t(n)In ∩In = ∅(compare [GJ]).Denote by L ⊂N the sequence of all levels m such that g◦t(m)c ∈Im−1∖Im which means that thereturn to the level m −1 is non-central. Let κ : N →N be the monotone surjective map such thatκ(m + 1) = κ(m) + 1 for m ∈L and κ(m + 1) = κ(m) otherwise.

The series of levels with the same κwill be called the cascades of central returns. (The “cascade” can degenerate to a single level if the centralreturn does not actually occur).

So, κ numbers subsequently the cascades of central returns.An important issue which we will discuss next is whether the sequence of grandkids shrink down to thecritical point. ( The main concern of this work will be the rate of shrinking).Unimodal renormalization.

Let us say that g ∈T ∞admits a unimodal renormalization if there existsa c -symmetric periodic interval.Then the return map to this interval is unimodal which justifies theterminology. It will be convenient to consider unimodal maps as “admitting unimodal renormalizations”.So, when we say that a map g ∈T ∞does not admit unimodal renormalizations, we assume automaticallythat it is not unimodal itself.

(The standard meaning of a renormalizable unimodal map corresponds toadmitting a unimodal renormalization with a period > 1 . )Lemma 2.3.

The following properties are equivalent:( i) g does not admit a unimodal renormalization. ( ii) The sequence of levels with non-central returns is infinite.

( iii) The grandkids In shrink to the critical point. ( iv) The set K(g) of non-escaping points has empty interior.Proof.

Let t(n) be as above the return time of c back to In−1 . Clearly, t(n) is monotonically increasing.

(i) ⇒(ii). Assume that the returns to all levels m ≥n−1 are central.

Then ∩m≥nIm is a c -symmetricg◦l -invariant interval. Contradiction.

(ii) ⇒(iii). If t(m) = t, m ≥n, eventually stabilizers then the returns to all levels ≥n −1 should becentral contradicting (ii).

If t(n) →∞but In don’t shrink to the critical point then the intersection ∩Inis a wandering interval. (iii) ⇒(iv).

Since an appropriate iterate of In covers the whole base interval I0 = J , there are escapingpoints in In . Since In shrink down to the critical point, there are escaping points in any neighborhood ofc .

But an appropriate iterate of any other interval must cover the critical point (no homtervals). Hence,escaping points are dense.

(iv) ⇒(i). Any periodic interval is contained in the filled-in Julia set.

⊔⊓The condition (iv) can also be stated in the following way:diam(J(n)(x)) →0asn →∞(2 −4)uniformly in x . If g is of finite type then it follows that K(g) is a Cantor set.Remarks.

1. One can also easily see that a map g ∈T admits a unimodal renormalization if and only ifthere is a an interval In−1 such that the critical point does not escape its kid In under iterates of f ◦t(n) .In the complex setting it will be accepted as the main definition.2.

One can also state and prove the Two kids Lemma by the same argument as in the complex setting (see§6). However, we don’t need it for the real discussion.Persistent and reluctant recurrence.Let B(x, ǫ) denote the interval centered at x of radius ǫ .Assume c is recurrent.

It is called reluctantly recurrent if there exist an ǫ > 0 and an arbitrary longbackward orbit ¯x = {x, x−1, ...x−l} in ω(c) such that the B(x, ǫ) allows a monotone pull-back along ¯x .Otherwise c is called persistently recurrent.6

Lemma 2.4. Assume g does not admit unimodal renormalizations.

Then the following two properties areequivalent:( R1) c is reluctantly recurrent;( R2) There is a critical puzzle-piece I(m)(c) with infinitely many kids.Proof. (R1) ⇒(R2) .

By (2-4) we can find a puzzle-piece J(n)(x) ⊂B(x, ǫ) . It is certainly a monotonepull-back of some critical puzzle-piece J(m)(c) .Let us pull J(n)(x) to depth n + l along the backward orbit ¯x .

Then let us consider the first momentwhen orb (c) crosses J(n+l)(x−l) and pull this puzzle-piece further to the critical point. We get a kid ofJ(m)(c) .

Letting l →∞we obtain infinitely many kids. (R2) ⇒(R1) .

Consider the pull-backs of J(m)(c) along the backward orbits of c creating the kids.⊔⊓Remark. Persistently recurrent situation appeared in [BL3, Lemma 11.1] and [GJ] under different names.In the complex setting it was introduced in [Y] under the name we use here.

The term “reluctantly recurrent”was suggested by McMullen.An invariant set K is called minimal if the orbits of all points x ∈K are dense in K .Lemma 2.5 (see [BL3], [Ma]). In the persistently recurrent case the critical set ω(c) is a minimal Cantorset.Proof.

Assume there exists an x ∈ω(c) whose orbit does not accumulate on c . Then for sufficiently smallǫ > 0 the pull-backs of B(g◦nx, ǫ) along the orb n(x) are monotone.

The contradiction proves that ω(c) isminimal. In particular, ω(c) does not contain periodic points.

Hence it is a Cantor set. ⊔⊓The minimality property yields that for any relative neighborhood U ⊂ω(c) all points x ∈U eventuallyreturn back to U , and, moreover, the return time is bounded (but certainly depends on U ).

So, on ω(c)the return map is of finite type: it is defined on the finitely many intervals. This motivates the followingconsideration.Return maps of finite type.

Take a nice interval L ∋c . Let K0 ∋c be its first kid.

Select also finitelymany other pairwise disjoint pull-backs Ki ⊂L, −n ≤i ≤m corresponding to the first returns of somepoints back to L . Then we can define the first return map of class T :h :m[j=−nKj →L.

(2 −5)In the most interesting case when L = I0 the combinatorial type of h can be described in terms of theg -itineraries of Ki through the intervals Ik of the previous level. To this end let us prepare a bit of algebraiclanguage.Spin semigroups.

Let Γ be a semi-group. We call Γ a spin semi-group if it is supplied with a characterǫ : Γ →Z2 .

If Γ is free then the spin structure can be certainly prescribed arbitrary on the generators, anduniquely determined by this data.Let p, q ∈N , and let us consider a free semi-groupΓ =< I−q, ..., I0, ..., Ip >with p + q + 1 generators which are linearly ordered with a marked generator I0 . If ǫ(I0) > 0 we say thatΓ is positively oriented.

Otherwise Γ is negatively oriented.Let W(Γ) ⊂Γ be the set of words γ = Ik(0)...Ik(s) such that the last symbol is I0 , while all others aredifferent from I0 . Let us order W(Γ) as follows (compare [MT]).

Let γt = Ik(0)...Ik(t) be the initial part ofγ , t = 0, ..., s . Assign to γ an integer vector with components ǫ(γt−1)k(t) , t = 1, ..., s .

The ordering ofΓ is induced by the lexicographic ordering of the corresponding vectors. Note that any two words of W(Γ)are comparable since they end with the I0 -symbol.So, the object we have described should be called a free ordered spin semigroup.

However, we willusually call it just “spin semigroup” keeping in mind the other structures.Let(Γ′, ǫ′)beanotherspinsemigroupofthesametypewithgeneratorsI′−q′, ..., I′0, ..., I′p′, and let χ : (Γ′, ǫ′) →(Γ, ǫ) be a semigroup homomorphism. Let us call it unimodal7

if χ(Ik) ∈W(Γ) and χ is unimodal on the set of generators, that is, it is strictly monotone on the setsI′−q′, ..., I′0 and I′0, ..., I′p′ , and has an extremum at I0 . Moreover, we require I0 to be the minimum ormaximum depending on whether Γ is positively or negatively oriented.A unimodal homomorphism χ is called admissible if additionallyǫ′(Kj) = sgnj ǫ(χ(Kj).

(2 −6)Return type. Take a map g ∈Tas in (2-1) of finite type.

Let the numbering of the intervals Ik beconsistent with the line order. Consider a free group Γg generated by Ik .

Let us supply Γg with a spinstructure ǫg . Spin of a non-central interval Ik is equal to +1 or -1 depending on whether g|Ik preservesor reverses orientation.

The spin of the central interval I0 is equal to +1 or -1 depending on whether c isthe minimum or maximum point.Assume now that the critical point returns to I0 . Then we can consider a return map (2-5):h :m[j=−nKj →I0of class T .

Let (Gh, ǫh) be an associated spin semigroup, and let us define a homomorphismχ : (Γh, ǫh) →(Γg, ǫg)assigning to each interval K = Kj its itinerary through the intervals of the previous levelχ(K) = Ik(1)...Ik(s−1) I0 ∈W(Γg)(2 −7)until the first return back to I0 , that is, gjK ⊂Ik(j) with k(j) ̸= 0 for j = 1, ..., s −1. Let us call theitinerary χ(K0) ∈W(Γg) of the central interval the kneading sequence of the return map.We call the homomorphism χ the type of the return map.

Observe that χ is admissible. Property(2-6) expresses the chain rule for the orientation type of the composition.

The unimodal property of χreflects the fact that the central branch of g is unimodal.Denote by T (q, p, ǫ) the subclass of Twith specified q, p ⊂N and spin character ǫ .Let gt ∈T (q, p, ǫ), t0 ≤t ≤t1 , be a continuous (in C0 topology) one-parameter family of maps.This meansthat the intervals Ik ≡Ik(t) move continuously with t , and each rescaled branch gt|Ik(t) ◦ψtk dependscontinuously on t (here ψtk : [0, 1] →Ik(t) is the orientation preserving rescaling). Let us call the familygt full if gt0I0(t0) covers all intervals Ik(t0) while gt1I0(t1) does not intersect the interiors of Ik(t1) (orvice versa).Lemma 2.6.

Let gt ∈T (q, p, ǫ), 0 ≤t ≤1 , be a full one-parameter family. Let χ : Γ′ →Γg be anadmissible homomorphism.

Then for some parameter value t ∈(0, 1) there is a return map ht ∈T whosereturn type is equal to χ . Moreover, there is a full family of return maps ht, t0 ≤t ≤t1, with the sameproperty.Proof.

Without loss of generality we can assume that the critical point is minimum for all gt , that is, Γg ispositively oriented. In what follows we omit t in notations keeping in mind that everything actually dependson it.

For every t let us consider the intervals Lj with itineraries χ(Kj) . Such intervals exist (becauseoutside the central interval g acts as the Bernoulli scheme), and continuously depend on t .

Moreover,g◦(lj−1)Lj = I0 where lj is the length of the itinerary χ(Kj) .Since g = gt is the full family, there is a parameter interval T = [t0, t1] such that the critical valuegc runs though the interval L0 from one boundary point to another as t runs through T . Since χ isunimodal (and Γg is positively oriented), L0 lies on the left of all other intervals Lj .

Hence for t ∈T wehave gI0 ⊃Li, i ̸= 0 .Let us now consider the pull-back Kj ⊂I0 of the intervals Lj by the central branch in such a way thatKj lies on the right of c if and only if j > 0 . Then the first return map h on ∪Kj has type χ .

Moreover,as t runs through T , the critical value hc runs all way through I0 , so that we have a full family. ⊔⊓Let g = g1 ∈T .

Assume that we can subsequently construct a sequence of return maps of class Tgn : ∪Ink →In−108

where I00 ⊃I10 ⊃.... is a sequence of the first kids. Let χn, n = 1, 2, ... , be the corresponding sequence ofreturn types.

The following statement says that these types can be combined independently. (“Do whateveryou want”.

)Lemma 2.7. Let gt ∈T (q, p, ǫ), 0 ≤t ≤1, be a full one-parameter family, Γ0 ≡Γg .

Let χn : Γn →Γn−1, n = 1, 2, ... be any sequence of admissible homomorphisms. Then there is a map g = gt which admitsa sequence of return maps gn of type χn .Proof.

By the previous lemma we can subsequently construct a nested sequence of parameter intervalsT1 ⊃T2 ⊃... in such a way that the return types to Tk are required up to the level n , while the retunmaps gtn+1, t ∈Tn , to the n th level form a full family. Intersecting these intervals we get a requiredparameter value.

⊔⊓In particular, we can fix the intervals Ik and consider the “standard” family of maps which are linearon the non-central intervals (and hence don’t depend on parameter) and quadratic on the central interval.Then by changing this central branch we can obtain a map of any type {χn} .The return graph. Let us consider a sequence...→Γ2→Γ1→Γ0χ3χ2χ1of admissible homomorphisms of spin semigroups.

We can associate to it the following graded graph ∆.Put on the n th level of ∆the generators {Ink } of Γn . Connect a vertex Ink with a vertex In−1jof theprevious level by l edges if In−1jhas multiplicity l in the word χn(Ink ) .

This graph contains the full“abelian” information about the sequence of homomorphisms.Observe that each vertex of level n is connected by a simple edge with the central vertex In−10of theprevious level. Let us call the graph (and the sequence {χn} ) irreducible if for any vertex Ink there is apath down to a central vertex In+s0(such that we go strictly downstairs along this path).Let us consider now a sequence of return maps gn .

Its type is given by a sequence of homomorphismsχn which provides us with a graph ∆. A vertex Ink is connected with In−1jby l edges if the gn−1 -orbitof Ink passes through In−1jl times before the first return back to the central interval In−10.

(Such a graphwas introduced by Marco Martens.) The following statement is immediate:Lemma 2.8.

The graph ∆is irreducible if and only if the critical orb( c ) crosses all intervals Ink . ⊔⊓Realization of all types.

Let g ∈T have a recurrent critical point such that f|ω(c) is minimal. Assumealso that g does not admit a unimodal renormalization.

Denote this class of maps by Tmin . Let us considerthe nested sequence of the first grandkids I0 ⊃I1 ⊃... .

Let us define a sequence of return mapsgn : ∪Ink →In−1,In ≡In0in such a way that we select only those intervals Inkwhich cover the critical set. The number of theseintervals is finite since ω(c) is minimal.We call gn a pre-renormalization of gn−1 .So, we obtain asequence ¯χg = {χn} of admissible homomorphisms with an irreducible graph ∆g .

Let us call this sequencea type of g .Lemma 2.9. Let gt ∈T (q, p, ǫ), 0 ≤t ≤1 be a full one-parameter family, Γ0 ≡Γg .

Let ¯χ = {χn :Γn →Γn−1}∞n=1 be any irreducible sequence of admissible homomorphisms such that χn(In) ̸= (In−1) forinfinitely many n . Then there is a map gt ∈Tmin of type ¯χ .Proof.

By Lemma 2.7 we know that there is a map gt which admits a sequence of return maps of typeχn . Since ¯χ is irreducible, this is a sequence of pre-renormalized maps (by Lemma 2.8).

Hence ¯χg = ¯χ .This map does not admit a unimodal renormalization since χn(In) ̸= (In−1) for infinitely many n .By Lemma 2.3 diam (In) →0 . Since the critical point returns to all central intervals In , it is recurrent.In order to see the minimality property, let us consider the union of orb( Ink ) until their first return back toIn−1 .

This provides us with a finite covering of ω(c) by the pull-backs of In−1 . It follows that the orbitof any point x ∈ω(c) crosses In−1 and hence ω(x) = ω(c) .

⊔⊓9

Combinatorial model of the critical set. Let Γm∗be a sub-semigroup of Γm generated by non-centralintervals Ik, k ̸= 0 .

Consider a space Ωof all finite and infinite sequences ¯δ = (δ1, δ2, ...) such thatδn ∈Γl(n)∗with a strictly increasing sequence of levels l(n) . Let us define a map σ : Ω→Ωin the followingfashion.

Set n = l(1) . If the word δ1 has length greater than 1 then just forget the first symbol of this word.If δ1 = (Ink ) and χ(δ1) = (In−10) then forget δ1 .

Finally, if δ1 has only one symbol and χ(δ1) ̸= (In−10)then replace δ1 with χ∗(δ1) where χ∗(δ1) coincides with χ(δ1) except for the last symbol In−10which isdropped.Given a map g ∈Tmin , we can associate to a point x ∈ω(c) an element of the space Ωin the followingway. Let x ∈In−1∖In .

Then consider the itinerary δ1 of x through the intervals of level n = l(1) untilthe first moment t when it lands in the central interval In . Then find a level l(2) = m > n = l(1) suchthat y = g◦tn x ∈Im∖Im−1 .

Coding now y in a similar way we will find δ2 , etc. Since diam(Ink ) →0 asn →∞, this provides us with a homeomorphism between ω(c) and Ωconjugating f and σ .

So, we havethe following lemma.Lemma 2.10. Two maps f, g ∈Tmin have the same type, ¯χf = ¯χg , if and only if their restrictions on thecritical sets are topologically conjugate by an orientation preserving homeomorphism respecting the criticalpoints.

⊔⊓Pull-back argument.Lemma 2.11. Two maps f and g of class Tmin are topologically conjugate by an orientation preservinghomeomorphism if and only they have the same types, ¯χf = ¯χg .Proof.

Let us set g ≡˜f and mark all objects related to g with the tilde. By Lemma 2.9, there is anorientation preserving conjugacy h : ω(c) →ω(˜c) putting c to ˜c .

Let us continue it to an orientationpreserving homeomorphism h : (J, ∪Ik) →( ˜J, ∪˜Ik) of the base intervals.Consider now a nested sequence of inverse images J(k) = f −kJ, k = 0, 1, ... shrinking down to the Juliaset K(f) , J(1) ≡∪Ik (and define˜J(k) similarly). Since h(c) = ˜c , we can lift h via the folding maps fand ˜f to an orientation preserving homeomorphism:h1J(1)→˜J(1)f ↓↓˜fJ→˜JhThen h1|∂J(1) = h , and hence h provides a continuation of h1 to the whole base interval J .Let us show that h1|ω(c) = h|ω(c) .

Indeed, since h is a conjugacy on ω(c) , the points h(x) and h1(x)either coincide or are ˜c -symmetric. Since both of maps are orientation preserving, they must coincide.Now let us repeat the construction and pull h1 back, etc.

In such a way we will construct a sequencehn : J →˜J of orientation preserving homeomorphisms such that(i) ˜f ◦hn = hn−1 ◦f ;(ii) hn agrees with h on the critical set;(iii) hn|J∖J(n) = hn−1 .Hence there is a pointwise limit H = lim hn which is an orientation preserving homeomorphismJ∖K(f) →˜J∖K( ˜f) conjugating f and˜f . Since the both Julia sets K(f) and K( ˜f) are Cantor, Hcan be automatically continued across these sets.

⊔⊓Example: the Fibonacci recurrence. In this case we have two intervals In0 and Inj , j ∈{−1, +1}, onall levels.

The returns are high on all levels, and χ(In0 ) = (In−1j, In−10) , χ(Inj ) = (In−10) . This data andthe spin structure on the first level (four possibilities) uniquely determine the admissible spin structures onthe further levels and the choice between j = 1 or −1 for the non-central intervals InjNamely, one cansee that on two subsequent levels the Injlie on one side of c , and then on the next two levels they lie onthe opposite side, etc (the first level can be the only exception).10

We see that there are four Fibonacci types. But actually two pairs of them are the same up to the choiceof the orientation of the base interval.

So, only two types are left which differ by the spin of the non-centralbranch. The renormalization interchanges these types.Let us remark in conclusion that the name “Fibonacci” comes from the observation that the first returntime tn of the critical point to the n th level satisfies the Fibonacci recurrent equation tn+1 = tn + tn−1 .Renormalization.

In the subsection “Realization of all types” we defined the pre-renormalization g1 ofa map g ∈Tmin . The renormalization Rg is obtained from g1 just by rescaling of the base interval In0(for g1 ) to the original interval J .

If we repeat this procedure, we will see that the n -fold renormalizationR◦ng is obtained from gn by rescaling In0 . The type of the renormalization is defined as the type of thecorresponding pre-renormalization.

Now we can summarize the above results as follows:Theorem 2.12. Let g ∈T .

Then g is infinitely renormalizable with R◦ng ∈Tmin . The topological typeof g is uniquely determined by the sequence ¯χg of renormalization types.

Any sequence of admissible types...→Γ2→Γ1→Γ0χ3χ2χ1can be realised in any full one-parameter family gt with Γg = Γ0 .Maps of infinite type. In the case when the critical point is recurrent but ω(c) is not minimal, we still canrenormalize the map on a nice interval by taking the first return map, keeping only the intervals intersectingω(c) and rescaling.

However, now the domain of the n -fold renormalization consists of infinitely manyintervals starting from some level. Still one can develop a similar combinatorial theory but we don’t need it.Let us only agree that In ≡In0 is still the central interval, and the intervals Ink with k > 0/k < 0 lie onthe right/left of c .Getting started.

Let f : L →K, L ⊂K, be a unimodal map, exact on [fc, f ◦2c] . Take a nice intervalJ ≡I0 ⊂[fc, f ◦2c] , and renormalize f on J .

We obtain a map g ∈T ∞which does not admit unimodalrenormalizations. This is our object to study.§3.

Estimates of Poincar´e lengths and scaling factors.Hyperbolic line and asymmetric Poincar´e length. Let us consider an interval L = [a, b] as a hyperbolicline with the Poincar´e metric 2dx/(x −a)(b −x) .

Let G be a subinterval of L , and U and Vbe thecomponents of L∖G (see Figure 1). We will use the following notations for the Poincar´e length of G in L :P(G) ≡P(G|L) ≡P(U, V ) = log(1 + |G||U|) + log(1 + |G||V |).Note that the bigger the space is around G in L , the smaller Poincar´e length P(G|L) is.UVGcLFigure 1Let us now state the main analytic tools of real one-dimensional dynamics (see [MS]).

They are called“the Schwarz Lemma” and “the Koebe Principle” by analogy with the classical facts in geometric functiontheory.Schwarz Lemma. Any diffeomorphism h : L →L′ with positive Schwarzian derivative contracts thePoincar´e metric.

⊔⊓Hence, given an interval G ⊂L and its image G′ = hG we have: P(G|L) ≤P(G′|L′) . So, if we havea definite space in L around G then we have also a definite space in L′ around G′ .11

Given a diffeomorphism h : I →I′ , let us callmaxx,y∈I log |f ′(x)||f ′(y)|its distortion or non-linearity. The case of zero non-linearity corresponds to linear maps.Koebe Principle.

Let h : (L, I) →(L′, I′) be a diffeomorphism with positive Schwarzian derivative,r = P(I|L) . Then the non-linearity of h on I is bounded by a constant C(r) independent of h .

Moreover,C(r) = O(r) as r →0 . ⊔⊓Lemma 3.1.

If L ⊃T ⊃I then P(I|L) ≤12P(I|T ) P(T |L).Proof. Since the Poincar´e metric is invariant under M¨obius transformations, we can normalize the intervalsin the following way: L = [−1, 1] , T = [−λ, λ], 0 < λ < 1 .

Let us consider the map h : x 7→λx . Thecalculation shows that ∥Dh(x)∥L ≤λ in the L -Poincar´e metric (with equality at 0).

On the other hand,h : L 7→T is an isometry from the L -Poincar´e metric to the T -one. Hence P(I|L) ≤λ P(h−1I|L) =λ P(I|T ).

Observe finally that 2λ ≤P(T |L) . ⊔⊓Remark.

We will actually use Lemma 3.1 in a slightly different form:P(I|L) ≤P(I|T ) P ∗(T |L)where P ∗(T |L) = min{P(T |L), 1} .Suppose we have a map g with a single non-degenerate critical point c . If the interval int G does notcontain c , then let us introduce the asymmetric Poincar´e length defined asQ(G) ≡Q(G|L) ≡Q(U, V ) = log(1 + |G||U|) + 12 log(1 + |G||V |),provided U is closer to c than V .

Clearly, Q(G) < P(G) . The coefficient 1/2 is related to the exponent2 of the critical point c .

It turns out that the asymmetric Poincar´e length behaves more regularly underrenormalizations of a quasi-quadratic map than the usual Poincar´e length.Parameters. From now on we will assume without change of notations that all maps of class Thavenegative Schwarzian derivative and a non-degenerate critical point c .

Let us consider a map g ≡q1 ∈Twhich does not admit unimodal renormalizations, and has a recurrent critical point. Then we can constructthe sequence of pre-renormalized mapsgn :[Ink →In−1with the central intervals In0 ≡In shrinking down to c .

In this section we don’t assume that ω(c) isminimal, so that we allow infinitely many intervals Ink on all sufficiently high levels n .By the gap between intervals U and V we mean the bounded connected component of R∖(U ∪V ) .Let us introduce the following parameters:- Kn is the infimum of asymmetric Poincar´e lengths Q(Ins , Int ) of the gaps between intervals of level nwith st ≥0 ;- µn = |In|/|In−1| is the scaling factor on level n ;-λn=maxi̸=0 [In−1:Ini ]is the maximal Poincar´e length of the non-central intervals oflevel n . Set λ∗n = min(λn, 1).Further, letαn = supk̸=0Inkdist(Ink , c)be a parameter which controls the non-linearity of the quadratic map φ(x) = (x −c)2 on non-centralintervals.By means of a C3 -small change of variable (near the critical value) we can make f purely quadraticin a neighborhood of the critical point c .

Then gn can be decomposed in the following way:gn|Ini = hn,i ◦φ,(3 −0)12

where φ(x) = (x−c)2 is the quadratic map and hn,i is a diffeomorphism with negative Schwarzian derivativeof an appropriate interval onto J . Let us consider one more parameter:- ρn is the maximal distortion of hn,i , and call it the distortion parameter.Let us remember that κ : N →N numbers the cascades of central returns (see §2).

The goal of thissection is to prove the following:Conditional version of Theorem B. There exist¯K and ¯µ > 0 (independent of a map) with thefollowing property.

If on some level N , KN > ¯K or µN < ¯µ then Kn →∞. Moreover, there existpositive constants A , C and σ such thatKn ≥A κ(n),µn+1 ≤C exp(−σκ(n)), n ∈L,ρn ≤C exp(−σκ(n)).Remark.

Observe that the scaling factors µn are exponentially small only on the special subsequence oflevels (outside the long cascades of central returns). However, the estimates of the Poincar´e lengths of thegaps, as well as the non-linearity control hold on all levels.Let us fix constants ¯µ and ¯ρ and¯K > 0 such that 1 > ¯µ > ¯ρ > 0 .

We will assume (until thesubsection “Cascades of central returns”) that the following estimates hold:ρn < ¯ρ,µn < ¯µ,Kn > ¯K. (3 −1)So, ¯ρ controls the distortion, ¯µ controls the scaling factors, and¯K controls the Poincar´e lengths ofthe gaps.

In what follows all constants depend on ¯µ (and actually on ¯ρ which becomes non-importantbecause we keep ¯µ > ¯ρ ) but not on the particular map. Sometimes we will abuse notations using the sameletter for different constants.Let us also fix small constants ¯λ and ¯α which separate range of small values of parameters λn andαn from big ones.Strategy.

Our strategy is the following. Let us consider two intervals U ′ = In+1kand V ′ = In+1jsuch thatthe gap G′ between them does not contain the critical point.

Let us push these intervals forward by gnuntil the first moment p when U = gpnU ′ ⊂Intand V = gpnV ′ ⊂Ins lie in different intervals of level n , thatis s ̸= t . Then loosely speaking, the Poincar´e length of the gap between U and V can be estimated frombelow by 2Kn+χ with an absolute constant χ > 0 .

Pulling this back by an almost quadratic map we get anestimate of the asymmetric Poincar´e length of the gap between U ′ and V ′ , namely Q(U, V ) ≥Kn + χ/2 .The argument depends on the positions of the intervals Intand Ins . The Fibonacci-like situation whenone of these intervals is central is the main one to look at (see Lemmas 3.3 and 3.9).

In all other cases theestimates are actually getting better.Estimates of P(U, V ) . This will occupy lemmas 3.2 through 3.7.Let us fix a level n and temporarily drop the index n in all notations so that In ≡I, gn ≡g, µn ≡µetc.

However, let In−1 ≡J . Let us take a non-central interval It, t ̸= 0 , of level n , and consider aninterval U ⊂It such that glU = I, giU ∩I = ∅, i = 0, 1, ..., l −1.

Sometimes we will write l = lU . Let Land R be the components of It∖U with L closer to c than R (see the following figure).UcLRI t13

Figure 2Let J+ and J−be the components of J∖I .Lemma 3.2. The following estimates hold:|U||L| + |U||R| ≤4µ(1 + O(µ)).

(3 −2)If l = 1 then|U||L| ≤2µ(1 + O(ρ + µ))(3 −3)andC(ρ, µ)−1 ≤|R||L| ≤C(ρ, µ),(3 −4)with C(ρ, µ) =√2(1 + O(ρ + µ)) . If l > 1 and then|U||L| ≤4µλ∗(1 + O(µ))and|U||R| ≤4µλ∗(1 + O(µ)).

(3 −5)Proof. There is an interval T such that U ⊂T ⊂It which is diffeomorphically mapped by gl onto J .By the Schwarz lemma,P(U|It) ≤P(U|T ) ≤P(I|J) = 4µ(1 + O(µ)).

(3 −6)ButP(U|It) = (|U||L| + |U||R|) (1 + O(P(U|It))),(3 −7)provided there is an a priori bound on [It : U] . The last two estimates imply (3-2) .In order to get (3-3) let us make use of the decomposition (3-0):|U||L| < |φU||φL| < |I0||J+|(1 + ρ) = 2µ(1 + O(ρ + µ)).Estimate (3-4) follows from the fact that g is the composition of a quasi-symmetric map φ and adiffeomorphism with distortion ρ (the√2 comes as the qs norm of φ−1 ).Suppose now that l > 1 .

Then gl−1 maps the interval T introduced above onto a non-central intervalIs, s ̸= 0 . Hence there is another interval T ′ in between T and It which is mapped by gl−1 onto J .Hence P(T |It) ≤P(T |T ′) ≤P(Is|J) ≤λ.

By Lemma 3.1 and estimate (3-6),P(U|It) ≤P(T |It)P(U|T ) ≤4µ min(λ, 1)(1 + O(µ)),(3 −8)and the estimates (3-5) follow from (3-7) and (3-8). ⊔⊓We will use the sign ≺or ≻if an estimate holds up to O(µ + ρ) , and a sign ≈if an equality holdsup to O(µ + ρ) (provided µ ≤¯µ, ρ ≤¯ρ ).Let U ⊂It be as above, H be the gap between I0 and It , G be the gap between I0 and U (seeFigure 3).

Let P(H) denote the Poincar´e length of H in I0 ∪H ∪It , and P(G) denote the Poincar´elength of G in I0 ∪G∪U . Notations Q(H) and Q(G) mean the asymmetric Poincar´e lengths of the samepairs of intervals.

Let J+ be the component of J∖I containing It .I0HItUGcFigure 3Lemma 3.3. If l = 1 then there is an absolute constant χ > 0 such thatP(G) ≻2Q(H) + χ.

(3 −10)14

If l > 1 thenP(G) ≻2Q(H) + log+ 1λ −log 2. (3 −11)Proof.

We haveP(G) ≥log(1 + |H||I| ) + log(1 + |H| + |L||U|) == log(1 + |H||I| ) + log(1 +|H||L| + |U|) + log(1 + |L||U|). (3 −12)The middle term is evidently bounded from below by log(1 + |H|/|It|) .

As to the last term, then by (3-2)we have:log(1 + |L||U|) ≻log 14µ ≈log(1 + |J+||I| ) −log 2 ≥≥log(1 + |H||I| ) + log |J+| + |I||H| + |I| −log 2,(3 −13)and the estimate P(G) ≻2Q(H) −log 2 follows.Let now l = 1 . Then we can use (3-3) instead of (3-2), and −log 2 disappears in the last estimate.We can also improve the estimate of the middle term of (3-12) as follows.

Because of (3-3) and (3-4), thereis a τ > 1 such that |It| ≥τ(|L| + |U|). Hencelog(1 +|H||L| + |U|) ≻log(1 + |H||It| ) + χ,(3 −14)provided H is not tiny as compared with It .

On the other hand if H is tiny as compared with It then J+is big as compared with H and H is big as compared with I (namely, log(|H|/|I|) ≥¯K−[a tiny term]).Hence the second termlog |J+| + |I||H| + |I|in (3-13) is big, and suppresses −log 2 . These yield (3-10).Finally, if l > 1 then we can improve (3-13) by using (3-5) instead of (3-2).

⊔⊓Now together with the above pair of intervals U ⊂It , let us consider a similar pair V ⊂Is, s ̸= 0,with V to be a monotone pull-back of I . Let us assume that both pairs lie on the same side of c , and thelatter one is closer to c than the former (see Figure 4).

Let G be the gap between V and U , H be thegap between Is and It .IHItUGcsVMNLRFigure 4Lemma 3.4. If Is and It are non-central intervals lying on the same side of c thenP(G) ≥P(H) + 2 log 1µ −O(1).

(3 −16)15

Proof. Let L and R be the components of It∖U as defined above, while M and N are the componentsof Is∖V .

Then we have:P(G) = log(1 + |H| + |N||V |) + log(1 + |H| + |L||U|) == log(1 +|H||N| + |V |) + log(1 + |N||V | )++ log(1 +|H||L| + |U|) + log(1 + |L||U|). (3 −18)The sum of the first and the third terms of (3-18) is certainly greater than P(H) .

Estimating the secondand the last terms by (3-2), we get (3-16). ⊔⊓The following lemma will allow us to handle the case when the non-linearity of φ is not small.Lemma 3.5.

The following estimate holds: µ = O((1 + 1α)e−K .Proof. Let us select an interval Ik for which α = |Ik|/dist(Ik, c).

Let W be the gap between I and Ik .Then we have:K ≤P(I, Ik) ≺log 12µ + log(1 + 1α),and the conclusion follows. ⊔⊓We will need the following lemma to analyze cascades of central returns.Lemma3.6.

UnderthecircumstancesofLemma3.4,ifH≥dist(H, c)thenP(G) ≥(5/2)K −O(1).Proof. Let W be the gap between I and Is and X be the gap between I and It .

Let us start withformula (3-18). Because of the assumption of the lemma, we can estimate from below half of its first termby (1/2) log(1 + |W|/|Is|) , and half of the third term by (1/2) log(1 + |X|/|It|) −log 2 .

The sum of theother halves we estimate as (1/2)P(H) .Remember that J+ denotes a component of J∖I . The second and the last terms we ≻estimate by(3-2) as log(1+|I|/|J+|) which is greater than both log(1+|I|/|W|) and log(1+|I|/|X|) .

Taking all thesetogether, we getP(G) ≥Q(I, Is) + Q(I, It) + 12P(H) −O(1) ≥(5/2)K −O(1).⊔⊓Finally, let us consider the case when Is and It lie on the opposite sides of c .Lemma 3.7. If Is and It lie on the opposite sides of c then P(G) ≻(5/2)K.Proof.

The argument is the same as in the previous lemma. The point is that now we automatically have|H| ≥|W| and |H| ≥|X| where as above W denotes the gap between I and Is , and X denotes the gapbetween I and It .

⊔⊓Quadratic pull-backs. Let us start with a lemma which says that the square root map divides the Poincar´elength at most by 2.Lemma 3.8.

Let us consider a quadratic map φ : x 7→(x −c)2 . Let U and V be two disjoint intervalslying on the same side of c , V being closer to c than U .

ThenQ(V, U) > 12P(φV, φU).Proof. We can assume that c = 0 and V , U lie on the right of c .

Let V = [v, a] , U = [b, u] . ThenP(V, U) −12P(φV, φU) == 12log a + va −v −log b + vb −v+ 12log u −au −b + log u + bu + a> 12 log u −au −b ,16

which is exactly what is claimed. ⊔⊓In Lemmas 3.3 - 3.7 we have estimated the Poincar´e length of the gap G between U and V .

Nowwe are going to use Lemma 3.8 in order to estimate the asymmetric Poincar´e length of the gap between U ′and V ′ . Again let us start with the situation when one of the intervals, say Is , is central (as in Lemma3.3).

As above, H denotes the gap between Is and It . Set T ′ = U ′ ∪G′ ∪V ′, T = gpT ′ = U ∪G ∪V.Lemma 3.9.

Under the circumstances just described there is a constant χ > 0 such thatQ(G′) ≻Q(H) + χ,(3 −22)orQ(G′) ≻K + 12 log+ 1λ −O(1). (3 −23)Proof.

Case 1. Let p = 1 .Let us use representation (3-0) of g|I as the quadratic map φ postcomposed by a diffeomorphism h withdistortion ρ .

Pulling G back by h and then by φ (making use of Lemma 3.8), we see that Q(G′) ≻(1/2)P(G) . Together with Lemma 3.3 this yields the claim.Case 2.

Let p > 1 .Then let us consider the intervals ˜U = gp−1U ′, ˜V = gp−1V ′ and ˜G = gp−1G′ . All three of them belong tothe same interval of level n , say Ij .Because of (3-0) we can consider the following decomposition:ψ ◦φ|T ′,(3 −25)where ψ is a diffeomorphism onto J .Remember that the non-linearity of the quadratic map φ|Ij iscontrolled by the quantifier α .

Let us take a small ¯α > 0 and consider several subcases.Subcase (i). Assume α < ¯α .This implies that g|Ik, k ̸= 0, is an expanding map with the expansion > 2 and small non-linearity.

Thenthe diffeomorphism ψ in (3-25) has small non-linearity as well. Together with Lemmas 3.3 and 3.8 thisyields the desired estimates.Let S be the gap between I and U .Subcase (ii).

Assume that |S| ≤¯α|J| .Then P(T |J) = O(¯α + µ) (make use of Lemma 3.2). Hence ψ|φT ′ has small non-linearity, and the resultfollows.Subcase (iii).

Finally, assume thatα > ¯α(3 −26)and|S||J| > ¯α. (3 −27)Then Lemma 3.5 and (3-26) imply|I||J| = µ = O(e−K).

(3 −28)Together with (3-27) this implies|I||S| = O(e−K). (3 −29)Let J−be the component of J∖I disjoint from S .

Given an interval X ⊂I , let˜X denote its pull-backby ψ . Pulling the interval J−∪I ∪S back by ψ , we get by (3-28), (3-29) and the Schwarz lemma|˜I|| ˜S|= O(e−K).

(3 −30)Let Q be the component of J∖U which does not contain c . ThenP(S, Q) = O(µλ∗) = O(λ∗e−K),17

and hence| ˜U|| ˜S|= O(λ∗e−K)(3 −31)as well. estimates (3-30) and (3-31) implyP(˜I, ˜U) ≥2K + log+ 1λ −O(1).

(3 −32)Pulling this back by the quadratic map, we get (3-23). ⊔⊓Lemma 3.10.

Let both intervals Is and It be non-central and lie on the same side of c . ThenQ(G′) ≻12K + log 1µ −O(1)(3 −33)orQ(G′) ≥K + B(K, λ) −O(1)(3 −34)where B(K, λ) ≥0 , and B(K, λ) = K/4 −O(λ) for λ ≤¯λ .Proof.

Let us again consider several cases depending on the non-linearity α of the quadratic map φon the non-central intervals. Let ¯α > 0 be small.Case 1.

Let p = 1 or α ≤¯α.For p = 1 let us use representation (3-0). For p > 1 the condition α ≤¯α holds.

Hence for any p ≥1gp|T ′ = ψ ◦φ|T ′(3 −36)where ψ is a diffeomorphism with bounded non-linearity. Hence pulling T back by ψ , we don’t spoil(3-16).

Composing this with φ -pull-back, we get (3-33) by Lemma 3.8.Case 2. Let p > 1 and α ≥¯α .

Then Lemma 3.5 yields µ = O(e−K) .Let W ⊃gT be a gp−1 -fold pull-back of J . Given an interval X ⊂J , denote by˜X ⊂W itsgp−1 -fold pull-back.

Pulling the pairs of intervals Is ⊃V and It ⊃U back to W we getP( ˜V : ˜Is) = O(µ) = O(e−K)andP( ˜U : ˜It) = O(µ) = O(e−K). (3 −37)Let us pull the interval Is ∪H ∪It back subsequently by g and then by gp−2 .

Apply Lemma 3.8 on thefirst step and the Koebe Principle on the second. This yieldsP(˜Is, ˜It) ≻12P(H) ≥2B(K, λ)(3 −38)where B(K, λ) is as was claimed.

Estimates (3-37) and (3-38) yieldP( ˜U, ˜V ) ≥2K + 2B(K, λ) −O(1).Pulling this back by g , we obtain (3-34). ⊔⊓Lemma 3.11.

Let Is and It be non-central intervals lying on the opposite sides of c . ThenQ(G′) ≥K + B(K, λ) −O(1)with B(K, λ) as in Lemma 3.10.Proof.

Let us again consider two cases.Case 1. Let p = 1 or α < ¯α .Then argue as in Case 1 of the previous lemma but use Lemma 3.7 instead of Lemma 3.4.

This yieldsQ(G′) ≥(5/4)K −O(1) which is better than what is claimed.Case 2. Let p > 1 and α ≥¯α.Then argue as in Case 2 of the previous lemma.

⊔⊓More relations between the parameters.Let us mark the quantifiers of level n + 1 by “prime”:µ′ ≡µn+1, λ′ ≡λn+1 etc. The following lemma provides us with rough estimates of the parameters of leveln + 1 through µ = µn .18

Lemma 3.12. The following estimates hold:λ′ = O(√µ)andµ′ = O(√µ),(3 −39)ρ′ = O(µ),(3 −40)K′ ≥12 log 1µ −O(1).

(3 −41)In the case of non-central return, (3-39) can be improved as follows:µ′ = O(pµλ∗). (3 −42)Proof.

Let us take an interval U ′ of level n + 1 and consider its image U = gU ′ ⊂It . If t = 0 we haveP(U|J) ≤P(It|J) ≈4µ.

Otherwise by Lemma 3.1 and estimate (3-2) we have P(U|J) = O(λ∗µ). Nowestimates (3-39) and (3-42) follow from decomposition (3-0).It follows from (3-0) and Lemma 2.2 that gn+1|U ′ = ψ ◦φ|U ′ where ψ is a diffeomorphism with theKoebe space spreading over J .

Since ψ(φU ′) ⊂I , (3-40) follows.In order to get (3-41) let us take two intervals U ′ and V ′ of level n + 1 and go through our basicconstruction (see the “Strategy”). Represent gp|T ′ as a composition ψ ◦φ where ψ is a diffeomorphismwith a Koebe space spreading over J .

Now pull the pair of intervals It ⊃U back by gp taking into accountthat P(U|It) ≺2µ . We see that |U ′|/|G′| = O(√µ) , and the result follows.

⊔⊓In what follows we restore the index n . Let us now treat the problem of estimating µn+1 through theparameters of lower levels.Lemma 3.13.

Let ǫ > 0 . In the non-central return case one of the following estimates holds:µn+1 = O(pλ∗n exp(−(1 −ǫ)Kn/2)),(3 −43)orµn+1 = O(exp(−(1 + ǫ/2)Kn/2)),(3 −44)orµn+1 = Oqµnµn−1λ∗n−1.

(3 −45)Proof. Let us again consider several cases.Case 1.

Let αn > exp(−ǫKn). Then by Lemma 3.5,µn = O(exp(−(1 −ǫ)Kn)).Using this and (3-42) we obtain (3-43).Case 2.

Let αn ≤exp(−ǫKn). Let U = gnIn+1 ⊂Int , t ̸= 0 .

Let L be the component of Int ∖U whichis closer to c and R be the other component.High return subcase. Then arguing as in Lemma 3.3 we see thatlog1µ2n+1≻log1 + dist(U, c)|U|≥log1 + dist(Int , c)|Int |+ log1 + |L||U|≥≥log12µn+ 12 log1 + dist(Int , c)|Int |+ 12 log1 + dist(Int , c)|Int |−O(1) ≥≥Kn + 12 log 1αn−O(1) ≥(1 + ǫ/2)Kn −O(1),and (3-44) follows.Low return subcase.Let b be the boundary point of In−1 lying on the same side of c as Int .SetX = [c, Int ], the convex hull of c and Int , Y = [Int , b] .

We need to refine the situation again. (i) Let |Y |2 ≥|X| · |Int | exp(ǫKn/2) .

Thenlog1µ2n+1≈log1 + dist(U, b)|U|≥log |Y ||Int |+ log1 + |R||U|≥≥12 log |X||Int |+ log12µn+ 12ǫKn −O(1) ≥(1 + ǫ/2)Kn −O(1),19

and we have (3-44) again. (ii) Let |Y |2 ≤|X| · |Int | exp(ǫKn/2) .

Then “an exponentially low return” occurs:|Y |/|In−1| ≤exp(−ǫKn/4).It follows thatdist(gn−1Int , gn−1b)/dist(gn−1Int , gn−1c) = O(exp(−ǫKn/8)). (3 −46)Let gn−1Int ⊂In−1j.

Then (3-46) implies that In−1jis a non-central interval, that is j ̸= 0 . Hence|gn−1Int |/|gn−1Y | = O(µn−1λ∗n−1).Since gn−1|Y has distortion O(exp(−ǫKn/4)) , we conclude that|Int |/|Y | = O(µn−1λ∗n−1)as well.

Together with P(U|Int ) = O(µn) this implies (3-45). ⊔⊓Now we are prepared to prove the Conditional Version of Theorem B.

To make life easier, let us firsttreat the case when there are no central returns at all.No central returns case. If µ0 is small then K1 is big by (3-41).

So, we can make the following inductiveassumption: There is a θ > 0 such thatKi ≥θi,i = 1, 2, ..., n,(An)andµi ≤12 exp(−θi/2),i = 1, 2, ..., n.(Bn)By (3-39) we haveλi ≤exp(−θi/4) << ¯λ,i = 1, 2, ..., n.(3 −47)Now Lemma 3.13 allows us to conclude that there is a δ > 0 such thatµi ≤exp(−(θ + δ)i/2),i = 1, 2, ..., n + 1. (3 −48)which is certainly stronger than (Bn+1) .In order to obtain (An+1) let us take a gap G′ between two intervals U ′ and V ′ of level n + 1 andpush it forward as described in the above “Strategy”.

Then we will find two intervals Insand Int . Let usconsider three cases depending on the position of these intervals.Case 1.

Let Ins be the central interval. Then by Lemma 3.9 and estimate (3-47) we conclude that thereis an absolute constant χ > 0 such that Q(G′) > Kn + χ which is greater than θ(n + 1) , provided θ wasselected to be smaller than χ .

Taking the infimum over all gaps G′ we obtain An+1 .Case 2. Let Insand Intbe two non-central intervals lying on the opposite sides of c .

Then Lemma3.10, assumption An and estimates (3-47), (3-48) give us a small δ > 0 such thatQ(G′) ≥(1 + δ)(n + 1)θ,(3 −49)which implies An+1 .Case 3. Let Insand Intbe two non-central intervals lying on the same side of c .

Then Lemma 3.11,the assumption (An) and (3-47) yield (3-49) again.Cascades of central returns. Let us have a non-central return on level m −1 followed by the cascade ofcentral returns on levels m, m+1, ..., m+q−1, and completed by a non-central return on level m+q,q ≥1 .So, m, m + q + 1 ∈L .

Set g = gm+1|Im+1 (see Figure 5). Theng(c) ∈Im+q∖Im+q+1, and gm+i|Im+i = g|Im+i, i = 1, ..., q + 1,(3 −50)and Im+i+1 is the g -pull-back of Im+i, i = 0, ..., q .

Let us call q the length of the cascade.20

Figure 5Let us fix a big natural number N . Let us define ω(m) in the following way.

If a non-central returnon level m −2 occurs, that is m −1 ∈L , then set ω(m) = 0 . Otherwise the level m −2 completes acascade of central returns of length p .

Then set ω(m) = min(p, N) .Let us assume by induction that there are θ > 0 and δ > 0 such thatKi+1 ≥((κ(i) + ω(i))θ,i ≤m, i ∈L(Am),andµi+1 ≤µ1 exp(−(θ + δ)κ(i)/2),i ≤m, i ∈L. (Bm)Our goal is to check (Am+q+1) and (Bm+q+1) , provided θ and δ are small enough.When we travel along the cascade of central returns the trouble is that the scaling factors µm+i isdefinitely increasing (and very fast: as (µm)1/2i ).

However, they are still quite small ( < ¯µ ) in the initialsegment of the cascade, so that we can apply all above lemmas. If µ1 is small enough then (Bm) guaranteesthat for i ≤Nµm+i ≤¯µ.

(3 −51)Moreover, both λn and αn are exponentially small, that is setting κ = κ(m) we haveλm+i = O(exp(−κθ/2))andαm+i = O(exp(−κθ)/2)), i = 2, ..., q + 1. (3 −52)Indeed, take a non-central interval Im+ikand push it forward by gi−1 .Since gi−1(c) ∈Im+1 whilegi−1Im+ik⊂Im∖Im+1 , there is a non-central interval Im+1tcontaining gi−1Im+ik.

Let X ⊃Im+ikbe thepull-back of Im+1tby gi−1 . Then X is contained in Im+i−1∖Im+i and P(Im+ik|X) = O(µm+1).

Thisestimate together with (Bm) implies (3-52).These considerations also show that gm+i can be represented as a composition of the quadratic mapφ and a diffeomorphism whose Koebe space is spread over Im . Hence, the distortion parameters remainsmall:ρm+i = O(exp(−κθ/2)),i = 2, ..., q + 1.

(3 −53)An estimate for Km+2 . A trouble with this estimate is that λm+1 need not be small.

However, by theinduction assumption and (3-39) the only way this can happen is if m −1 ̸∈L and m −2 completes a longcascade of central returns, that is ω(m) = N is big which makes the assumption (Am) stronger.More specifically, let us follow the above “Strategy”. Take a gap G′ between two intervals of level m+2and push it forward by iterates of gm+1 until its endpoints are separated by different intervals Im+1sandIm+1tof level m + 1 .

As usual, let us consider several cases depending on the positions of these intervals.Case 1. Let Im+1sbe central.

Then as we have explained either λm+1 < ¯λ orKm+1 ≥κθ + A(3 −54)with a big A . In both cases Lemma 3.9 yieldsQ(G′) > (κ + 1)θ(3 −55),provided θ is small enough.Case 2.

Let Im+1sand Im+1tbe non-central lying on the same side of c . If λm+1 < ¯λ then Assumptions(Am) , (Bm) and Lemma 3.10 imply that there is an ǫ > 0 such thatQ(G′) > (1 + ǫ)κθ(3 −56),which is certainly better than (3-55).If λ ≥¯λ then (3-54) holds.

Together with Lemma 3.10 this yields (3-55).Case 3. Let Im+1sand Im+1tbe non-central intervals lying on the opposite sides of c .

Then argue as inthe previous case using Lemma 3.11 instead of 3.10.So, in all cases (3-55) holds, and hence Km+2 ≥θ(κ + 1) .21

Estimates for Km+i+1, i > 1 , in the initial segment of the cascade, (while (3-51) holds). Now λm+1 isexponentially small by (3-52) but µm+i need not be exponentially small.

Let us assume by induction thatKm+j+1 ≥(κ + j)θ, j = 1, ..., i −1. (3 −57)To pass to the next level let us apply again our strategy and go through the same bunch of cases dependingon the positions of Im+isand Im+it.

Cases 1,2,3 mean the same as above.Case 1. Then Lemma 3.9 and (3-52), (3-57) yieldQ(G′) ≥(κ + i)θ.

(3 −58)Case 2. Let H be the gap between Im+isand Im+it.

If |H| ≥dist(H, c) then Lemmas 3.6 and 3.8 yieldthe desired estimate. Otherwise g has a bounded distortion on T ′ = U ′ ∪G′ ∪V ′ (the notations are thesame as in the Strategy description), and it follows from Lemma 3.4 thatP(G′) > Km+i + 2 log 1¯µ −O(1) > Km+i + χ,provided ¯µ is small enough.Case 3 is treated in the standard way using Lemma 3.11 and (3-52).Conclusion: (3-57) follows for j = i .Distortion control in the tail of the cascade.

Let M be the first moment for which µm+M ≥¯µ. Sinceµm+k+1 ≈√µm+k for k ≤q ,|Im+q+1| ≥c|IM|(3 −59)(with c ≈¯µ2 ).Lemma3.14.

Themapg(j+1)hasboundeddistortiononbothcomponentsofIM+j∖IM+j+1 , 0 ≤j ≤q .Proof. As we know g is almost quadratic.

Hence by (3-59) it has bounded non-linearity ng on IM∖Im+q+1 .Let L be a component of IM+j∖IM+j+1 . Then f kL is a component of IM+j−k∖IM+j+1−k .

Hence bythe standard argument the non-linearity of gj+1 on L is bounded byngjXk=0|gkL| ≤ng|IM∖Im+q+1| = O(1).⊔⊓Pulling now the intervals from level M back to the tail of the cascade, we conclude thatKM+i ≥KM −O(1) ≥(κ + M)θ,i ≤q −M + 1(3 −60)(for, perhaps a bit smaller θ ).A Markov scheme. Let us build up a Markov map F .

Let N be a big number as selected above. SetKm+2Nj= Im+2Nj, j ̸= 0 , and pull these intervals back by iterates of g to levels m+i, i = 2N, ..., m+q+1.Denote the corresponding intervals by Km+ij.

Now setF|Km+ij= gfor i > 2NandF|Km+2Nj= gm+2N.This map F carries Km+ijonto Km+i−1jfor i > 2N , and carries Km+2Njonto Im+2N−1 covering allintervals of our scheme.Proof of (Am+q+1) . Take two intervals U ′ = Im+q+2kand V ′ = Im+q+2lof level m+ q + 2 , consider theirimages by g , and then push them forward by iterates of F until the first moment they don’t belong to thesame interval Km+ijof our scheme.

This long-term composition is almost quadratic as one can see from(3-52). Denote the corresponding images of U ′ and V ′ by U and V .

We again have to consider severalcases.Case 1. Assume that for some N ≤j ≤q + 1Vbelongs to a central interval Im+j while V ⊂Im+jt, t ̸= 0 .

If j > M push these intervals forward to level M . By Lemma 3.14 this results in a bounded22

change of the Poincar´e length of the gap between the central interval Im+j and V . Let us denote newintervals by ˜U and ˜V .

Now they lie on the level l = min(M, j) . Then Lemma 3.3, estimate (3-52) andthe above estimates of Km+i yieldP(Il, V ) ≥2Km+2N ≥2(m + N)θ.Pulling this back by the quadratic map postcomposed with a bounded distortion map, we get the desiredestimate.Case 2.

Let V ⊂Im+js, U ⊂Im+jtwith s ̸= 0 and t ̸= 0 . As in the previous case, pushing theseintervals forward, we can assume that j ≥M .

Let H be the gap between Im+jsand Im+jt.Subcase (i). Let H ∋c .

The the standard argument based on Lemma 3.7 gives the desired estimate.Subcase (ii). Let H ̸∋c and |H| ≥dist(H, c) .

Use Lemma 3.6 instead of 3.7.Subcase (iii). H ̸∋c and |H| < dist(H, c) .

Then let us push the intervals forward by iterated g . SetUn = gnU etc.

Assume there is a moment for which |Hn| ≥dist(H, c) . Then gn|U ∪H ∪V has a boundeddistortion for the first such moment.

Hence we can argue as in the previous subcase.If there is no such a moment, then push the interval to the very beginning of the cascade (to level m )and apply Lemma 3.4.Proof of (Bm+q+1) . We should estimate µm+q+2 .

Let us push Im+q+2 forward to higher levels:T l = GlIm+q+2 ≡gl ◦gl+1 ◦... ◦gm+q+1Im+q+2.Let us stop on the highest level l < M for which one of the following properties hold:(i) the map Gl is not exponentially low in the sense of Lemma 3.13, Case 2-ii (that is GlIm+q+1 belongsto the exp(−ǫKl−1)|Il−1| -neighborhood of ∂Il−1 ), or(ii) l −1 ∈L and κ(l −1) = κ(m) −1 . This means that we have arrived at the beginning of the previouscascade.It follows from Lemma 3.14 that G is a quadratic map postcomposed with a bounded distortion map.This allows us to apply Lemma 3.13 for Gl instead of g , and to estimate µm+q+2 through the parametersof level l .

If (i) occurs then (3-43) or (3-44)-like estimates hold. Together with the above estimates of Kland λl they yield the desired estimate.Otherwise there is a non-central interval Ils ⊃Tl which can be monotonically pulled back by Gl .

SinceP(Tl|Ils) = O(µlµl−1...µm+q+1) = O(exp(−5θ/4)),we conclude thatµm+q+2 = OqP(Tl|Ils)= O(exp(−5θ/8)),and (Bm+q+1) follows.§4. Real bounds and limits of renormalized maps.In this section we will prove Theorem B for maps with a non-minimal critical set ω(c) and for mapsof unbounded type.

For maps of bounded type will show that if the scaling factors stay away from zerothen the family of renormalized maps is compact, and all limit maps are real analytic. As usual, let us firstassume that there are no central returns.

Set xm = f mx .A priori bounds. As in §3 let us consider the decomposition (3-0) gn|Ini = hn,i ◦φ , and denote by ρnthe maximal distortion of the diffeomorphisms hn,i .Theorem 4.1 (Martens [Ma]).

The distortions ρn are uniformly bounded.Lemma 4.2. The scaling factors µn = |In|/|In−1| are bounded away from 1.Proof.

The case of low return on level n −1 was treated in ([Ma], Lemma 3.7).In the case of high return let us assume that µn is close to 1. Then because of the bounds given byTheorem 4.1, the next scaling factor µn+1 = O(√1 −µn) will be very small.

Then by the results of §3µm →0 . ⊔⊓23

Lemma 4.3. All Poincar´e lengths P(Inj |In−1) are bounded away from ∞.Proof.

This follows from the previous lemma and Lemma 3.12. ⊔⊓Let ¯λ denote an upper bound of Poincar´e lengths P(Inj |In−1) .

Consider two intervals T ⊃G with Land R to be the components of T ∖G . Denote by¯σ =e¯λ1 + e¯λ < 1(4 −1)an upper bound of |G|/(|G| + |L|) provided P(G|T ) ≤¯λ .Orders and ranks.

Let us define the order ord n0 of the return to level n as the return time of gn -orb( c )back to In . Let us also define l -orders ord nlas the return time of gn−l -orb( c ) back to In .

In terms ofthe return graph the ord n0 is just the number of edges beginning at In+1 (and leading to the previous leveln ). The ord nlis the number of paths of length l + 1 beginning at In+1 (and leading to the level n −l ).Lemma 4.4.

If the scaling factors µn stay away from 0 then for each l the l -orders ord nlof returns toall levels are uniformly bounded.Proof. If on level n a return of high order p occurs then by Lemma 4.3 the next scaling factor µn+1 =O(√¯λp) is very small.

Similarly, for a given l , if ord nl is big then traveling down the graph from level n−lto n + 1 we see that µn+1 is small as well. ⊔⊓Let us consider now the Markov family M of intervals obtained by pull-backs of the initial interval I0(see §2).Let us assign to the critical intervals K ∈M rank 0.

Let us say that an interval K ∈M, K ⊂In−1∖Inhas rank k, k ≥1 , if orb( c) passes through it before the first return to In+k−1 but after the first returnto In+k−2 . For example, k = 1 if orb( c ) passes through K before the first return to In .

For K = Injthis can be nicely expressed in terms of the return graph as the length k of the shortest path leading fromInjdown to a central interval In+k .Lemma 4.5. Let K ∈M, K ⊂In−1∖In , and rank (K) ≥k > 0 .

Let us take a point x ∈In+k−2∖In+k−1 ,and consider the first moment l when f lx ∈K (provided there is one). Then the interval K can bediffeomorphically pulled back along the orb l(x) to an interval K′ ⊂In+k−2∖In+k−1 .Remark.

We don’t claim that K′ ∋x but just some iterate xs, s ≤l, so that the pull-back has lengthl −s .Proof. If k = 1 there is nothing to prove (set K′ = K ).

If k > 1 let us consider the last moment s < lwhen the orb (x) visits In . Then there is a moment p such that g◦pn (xs) ∈K and all intermediate iteratesg◦mn (xs) ̸∈In, 0 < m < p .

I claim that the pull-back of K along the gn -orbit of xs is monotone. Indeed,otherwise g◦pn (c) ∈K while g◦mn (c) ̸∈In, 0 < m < p , so that rank (K) = 1 .Let K1 ∋xs be the monotone pull-back of K along the gn -orbit of xs .

Then K1 ⊂In∖In+1 andrank (K1) ≥k −1 . So, we can proceed by induction.

⊔⊓Lemma 4.6. If the scaling factors µn stay away from 0 then ranks of all intervals Injare uniformlybounded.Proof.

Let rank Inj = k . Let l be the first moment when orb( c ) visits Inj .

Then by the definition of rankthere is an s < l such that cs ∈In+k−2∖In+k−1 . By the previous lemma we can monotonically pull Injtothe level n + k −2 along the orb cs .

We obtain an interval K′ ⊂In+k−2∖In+k−1 .Let us consider the interval In+k−1icontaining cs .Then one can see by induction (involving theSchwarz lemma) thatP(In+k−1i|In+k−2) ≤P(In+k−1i|K′) ≤¯σk−1with ¯σ from (4-1). Since orb( c ) passes through In+k−1ibefore its return to In+k−1 , the scaling factorµn+k = O(¯σ(k−1)/2) is small if k is big.

Contradiction. ⊔⊓Non-minimal case.

Now we are ready to proof Theorem B in the non-minimal case.24

Lemma 4.7. Assume that the critical set ω(c) is not minimal.

Then the scaling factors µn go down to 0,n ∈L .Proof. Indeed, in the non-minimal case there is a level n and a point x ∈ω(c) ∩In−1 which never passesthrough the central interval In .

It follows that the return time of points of orb (c) back to In is unboundedand hence there are infinitely many intervals Inkof level n . The ranks of these intervals certainly mustgrow up to ∞.

Now lemma 4.6 provides us with the starting condition for Theorem 3.0. ⊔⊓Unbounded Combinatorics.

Assume that ω(c) is minimal. Let us say that f is of bounded type if thenumber of intervals on all levels is uniformly bounded, and of unbounded type otherwise.

The unboundedcase can also be treated by a purely real argument (I owe this remark to Swiatek).Lemma 4.8. If f has unbounded combinatorics then the scaling factors µn go down to 0, n ∈L .Proof.

If the combinatorics are unbounded then either the l -orders of returns or ranks of the intervalsare unbounded (consider the return graph from §2). Now the required starting condition for Theorem 3.0follows from lemmas 4.4. and 4.6.

⊔⊓Bounded Combinatorics. This is the main case when we need to involve complex analytic methods.In this subsection we will show that provided the scaling factors stay away from 0, there is a sequence ofrenormalized maps C1 -converging to an analytic map.Here by the gaps of level n we will mean the components of In−1∖∪Inj .Lemma 4.9.

If the scaling factors stay away from 0 then all intervals and all gaps of level n are commen-surable with In−1 .Proof. Let us show first that the intervals Inimay not be tiny as compared with In−1 .

Indeed, because ofLemma 4.3 such an interval should lie very close to ∂In−1 . On the other hand, it follows from Theorem 4.1that gn−1 is quasi-symmetric.

Hence, gn−1Injlies very close to ∂In−2 and, moreover, gn−1In−1 coversthe interval In−1s⊃gn−1Inj(again because of Lemma 4.3). Hence we can monotonically pull In−1sback bygn−1 .Now we can apply the same argument to the interval In−1sand map gn−2 and so on.

In such a waywe will find a big l and an interval In−ltwhich can be monotonically pulled back by gn−l ◦... ◦gn−1 alongthe orbit of Ini . Let K ⊂In−1 be this pull-back.

This interval provides us with a big space around Ini ,namely |Ini |/|K| ≤¯σl . Since rank Iniis bounded by Lemma 4.6, we can pull this space back and obtain asmall scaling factor.

Contradiction.Let us now consider the gaps. They may not be too big as compared with the intervals since otherwisethe intervals would be tiny as compared with In−1 .

Let us consider any gap G in between Iniand Inj .Arguing as in the proof of estimate (3-41) one can see that the Poincar´e length P(G) is bounded from belowprovided the scaling factors are bounded away from 1. This means that G is not tiny as compared with oneof the intervals Inj , Ini .

Since these intervals are commensurable with In−1 , so the interval G is as well.As to the two “boundary” gaps, they are not too small as compared with the attached intervals becauseof Lemma 4.3. ⊔⊓Consequently, we can select a sequence of renormalized maps Rnf in such a way that the configurationsof intervals and gaps converge to a non-degenerate configuration of intervals and gaps.

Use now the rescaledrepresentation (3-0) for these maps:Rnf | ˜Ini = Gn,i ◦φwhere ˜Ini are the rescaled intervals of level n and Gn,i are diffeomorphisms of appropriate intervals onto theunit interval. By Theorem 4.1 and the previous lemma, the inverse maps G−1n,i have uniformly bounded C2 -norms, and hence form a C1 -compact family.

So, we can select a C1 -convergent sequence of renormalizedmaps,G−1n(s),i →G−1iasn(s) →∞.Each Gi is a long composition of the square root maps and diffeomorphisms whose total distortion iscontrolled byωn =Xjp(n,j)−1Xm=1|f m(Inj )|25

where p(n, j) is the return time of Injback to In−1 . But ωn →|ω(c)| = 0 by [Ma].

Hence the totaldistortion of the diffeomorphisms involved is vanishing.Now the “Shuffling Lemma” (see [S] or [MS], Ch. VI,Theorem 2.3) yields that the limit of renormalizedmaps is real analytic and, moreover, belongs to so called Epstein class which we are going to study in thenext section.Cascades of central returns.

Let us have (as in the end of §3) a non-central return on level m −1followed by the cascade of central returns on levels m, m = 1, ..., m + q −1 , and completed by a non-centralreturn on level m + q . The following remarks allow to adjust the previous analysis to this case.First of all, the first scaling factor µm of the cascade stays away from 1 by lemma 4.2.Assume now that the starting conditions don’t hold, that is, the scaling factors stay away from 0.

Thenµm+q stay away from 1. For, otherwise we have a high return on the level m + q −1 , and the next scalingfactor µm+q+1 is tiny (see the argument of Lemma 4.2).Furthermore, the ratio|Im+q||Im|≥δ > 0(4 −2)also stays away from 0.

For, otherwise the non-central intervals Im+qkhave a small Poincar´e length in theappropriate component of Im+q−1∖Im+q . This would enforce µm+q+1 to be small again.Consequently the mapg◦(q−1)m: Im+q−1∖Im+q →Im∖Im−1has a bounded distortion.

Indeed, since gm is quadratic up to a bounded distortion, by (4-2) it has boundednon-linearity on Im∖Im+q . Since the iterates of Im+q−1∖Im+q are disjoint, the claim follows.Now we can consider the return graph skipping all intermediate levels between m + q −1 and m , andto define the orders and ranks of the intervals through this graph.

As we have shown, on all levels of thegraph we have a priori bounds of the scaling factors, and passage from one level to another has a boundeddistortion. Now we can repeat the above argument.§5.

Epstein class and complex bounds.The goal of this section is to show that an appropriate renormalization of an analytic map of Epsteinclass is polynomial-like.Polynomial-like maps. By a polynomial like map we mean an analytic branched coveringf : ∪li=0Ui →Vwhere Ui and Vare topological disks, cl Ui ⊂V .

Let us consider a class A of polynomial-like maps fhaving a single non-degenerate critical point c ∈U0 .Epstein class. Given an interval I ⊂R , let P(I) = int (C∖(R∖I)) denote the complex plane slittedalong two rays, D(I) denote the disk based upon I as a diameter.

Let g be a real analytic map ∪Ik →Jsatisfying the following properties:( i). For k ̸= 0 there is the inverse map (g|Ik)−1 which univalently maps P(J) onto P(Ik) .

( ii). g|I0 = h ◦φ , φ(z) = (z −c)2 is the quadratic map, and h−1 has a univalent analytic continuationto P(J) .Let us call this class of maps Epstein class E (compare [S]).

Let us start with a general lemma fromhyperbolic geometry which was an ingredient of Sullivan’s Sector Lemma. It is essential for our complexbounds as well.Lemma 5.1.

Let φ : P(I) →P(J) be an analytic map which maps I diffeomorphically onto J . ThenφD(I) ⊂D(J).Proof.

The interval I is a Poincar´e geodesic in P(I) , and the disk D(I) is its Poincar´e neighborhood (ofradius independent of I ). Since φ contracts the Poincar´e metric, we are done.

⊔⊓High returns. In order to make the following discussion more comprehensible let us dwell first on the casewhen high returns occur on all levels (compare [LM], §8).26

Lemma 5.2. Let f ∈E .

Assume that we have high returns on all levels. Then gn is polynomial-like forsome n .Proof.

We can assume that the scaling factors stay away from zero. Then given an arbitrary small σ > 0 ,we can select a moment n such thatµn ≥µn−1(1 −σ).

(5 −1)Set q = gnc ∈In−1∖In, Im = [αm, βm] where βm lies on the same side of c as q (see Figure 6).Let us estimate the Poincar´e length P0 of [q, βn−1] in [αn−1, βn−2] :P0 ≤log1 + 1 −µn1 + µn+ log1 + µn−1(1 −µn)1 −µn−1≤log21 + µn+ log(1 + µn) + o(1) = log 2 + o(1)asσ →0(5 −2)(we have replaced µn−1/(1 −µn−1) by µn/(1 −µn) using (5-1) and monotonicity of the linear-fractionalfunction x/(1 −x), x ∈(0, 1) , and boundedness of µn away form 1 (Lemma 4.2)). Let us consider now therepresentation gn = h ◦φ where φ is the quadratic map and h is a diffeomorphism of an interval T ⊃φInonto [αn−1, βn−2] .

Set T = φIn ∪M ∪R where M and R are h -pull-backs of [q, βn−1] and [βn−1, βn−2]respectively. By the Schwarz lemma and (5-2)log1 + |M||φIn|≤P0 ≤log 2 + o(1),hence|M||φIn| ≤1 + o(1)asσ →0.

(5 −3)Figure 6Let us now take the set V = D(In−1) and pull it back by h . By Lemma 5.1 we will obtain a domainU ′ contained in D(φIn ∪M) .

Pulling this domain back by the quadratic map φ we obtain by (5-3) aconvex domain U0 which is almost contained in the disk D(In) .On the other hand, pulling V back by the univalent branches of gn , we will get by Lemma 5.1 domainsUk such that Uk ⊂D(Ink ),k ̸= 0 . We conclude that the sets cl( Uk ) are pairwise disjoint and are containedin V .

⊔⊓Cut-offiterates. Let us define now a “cut-off” iterate g◦ln K of an interval K ∋c inductively in thefollowing way:g◦ln K = gn(g◦(l−1)nK ∩Ijn)where Inj ∋g◦(l−1)nc .

If K = In then one boundary point of its cut-offiterates belongs to the ∂In−1 . Letus select the first moment l for which g◦ln (In, c) ∩In ̸= ∅.

In the low return case l > 1 .Low returns (a particular case).27

Lemma 5.3. Let us select a level n for which (5-1) holds, and find an l as described above.

Assume thatg◦ln c̸∈In .Thenappropriatepull-backsofD(In−1)formapolynomial-like map.Proof. Denote q = g◦ln c , p = g◦(l−1)nc , Im = [αm, βm] where βm lies on the same side of c as q .

Let Inibe the interval of level n containing p , and let p divides it into the intervals L and G with G closer toc than L . Note that gnL = [αn−1, q] , gnG = [q, βn−1] .Let us consider representation (3-0).

Since hn,i has a Koebe space spread over In−2 , one can get thefollowing estimate in the same way as (5-3):|φG||φL| ≤1 + o(1)asσ →0. (5 −4)Set µn = µ .

Since the non-linearity of φ on Iniis at most log(1/µ) , we obtain the following estimate:|G||L| ≤1µ (1 + o(1))asσ →0. (5 −5)Let R be the component of In−1∖Inicontaining the critical point c .

Let us estimate now the Poincar´elength P(L, R) . It follows from (5-5) that|G| ≤|Ini |1 + µ (1 + o(1)) ≤1 −µ1 + µ |In−1| (1 + o(1)).

(5 −6)HenceP(L, R) ≤log1 + |G||L|+ log1 + |G||R|+ o(1) ≤≤log1 + 1µ+ log1 +1 −µ(1 + µ)2+ o(1) == log1 +2µ2 + µ+ o(1) < log1 + 1µ2+ o(1). (5 −7)Let T ∋c be the pull-back of In−1 by gln .

Then gl−1nT = L . Decompose gl−1n|T = h ◦φ whereh : φT ∪G′ ∪R′ →L ∪G ∪R = In−1is a diffeomorphism.

By the Schwarz lemma and (5-7),|G′||φT | ≤1µ2 (1 + o(1)).Set ν = |T |/|In−1|. It follows from the a priori bounds of §4 that ν/µ = |T |/|In| stays away from 1.Hence the last estimate can be rewritten as|G′||φT | ≤τ 2ν2(5 −8)with an absolute constant τ < 1 .Let us consider now the disk D(In−1) .

By Lemma 5.1 its pull-back V ′0 by h is contained in the diskD(φT ) . Pulling V ′ by the quadratic map φ we obtain a domain V0 based upon the interval T .

By (5-8)the outer radius of V0 around c is less than (τ/ν)|T | = τ|In−1| . Hence cl (V0) ⊂D(In−1) .We have completed the construction of the central domain V0 .Let us now construct non-centraldomains of definition of our polynomial-like map.

First of all, take a non-central interval Iniof level nand pull the disk D(In−1) back by the corresponding branch of g−1n. By Lemma 5.1 we obtain a domainW ni ⊂Ini .Let now x ∈In∖T .

Then there is a moment s < l when gn -orb (x) is separated from gn -orbit ofT by intervals of level n . In other words g◦sn x ∈Injwhile g◦sn T ∩Inj = ∅.

Hence we can monotonicallypull Inj back to x by g◦snand obtain an interval T (x) . Moreover, we can univalently pull the domain W njback to x and obtain a domain W(x) ⊂D(T (x)) based upon T (x) .

A map g◦(s+1)nunivalently carriesthis domain onto Dn−1 .28

Let us consider the whole (finite) bunch of intervals Iniand T (x) , and the corresponding bunch ofdomains W niand W(x) . Let us redenote them as Ti and Vi, i = 1, ... respectively.

Since Vi ⊂D(Ti) ,these domains are pairwise disjoint. They are also disjoint from the central domain V0 .

Indeed, the domainsφVi, i = 0, 1, ... are pairwise disjoint since they are contained in the disks based upon disjoint intervals.So, we have a polynomial-like map H : ∪Vi →D(In−1) . ⊔⊓A remark on scaling factors.

The map H constructed above satisfies the following property: the gn -first return map to T0 coincides with the H -first return map to T0 . Indeed, if H|Ti = g◦mn , i ̸= 0, thenby looking through the construction we see that g◦kn Ti does not intersect T0 .Let T0 ≡T 0 ⊃T 1 ⊃... be the sequence of the central intervals of the renormalized maps R◦NH , andνn be the corresponding sequence of the scaling factors.

Then the above property of the first return mapsyield that there is an N such thatIn+N ⊂T n ⊂In+N−1.Hence the scaling factors νn can be estimated through the µn and vice versa. It follow that νn →0 if andonly if µn →0 .Low returns: a general construction.

The particular construction described above will be one step ofthe general construction. Let us start with a maph1 : ∪T 1,0i→T 0of class A .Let us order pairs (n, j) of integer numbers lexicographically.We will construct a finitehierarchical family of intervals T n,ji, (n, j) ∈E ⊂Z2, satisfying the following properties:0) E = {(n, j) : 0 ≤n ≤N, 0 ≤j ≤j(n)} , T 0 ≡T 0,00, T n,j0≡T n,j are symmetric intervals containingc ;Let (n′, j′) ∈E be the lexicographic successor of (n, j) ∈E .

Then1) T n,j ⊃cl (T n′,j′i) ;2) There is a map hn : ∪T n,0i→T 0 of class A induced by h1 ;3) For j > 0 the intervals T n,jiare obtained from T n,0iby pulling back by the central branch of hn .Moreover, hnc ∈T n,j for all j except for the last one j(n) (the reader can recognize here the cascades ofcentral returns).4) For n < N , hnT n,0 ̸∋c , while hNT N,0 ∋c .Under such circumstances we will also consider the mapHn :[m

If hnc ∈T n,0 and hnT n,0 ∋c , we consider the cascade of central returns until thefirst moment of high return. It produces the intervals T n,jiby pulling T n,0iback by the central branch ofhn .

Then we stop.Low case. Acting as in the above particular case let us consider cut-offiterates H◦mn T n,0 of the centralinterval until the first moment l whenH◦ln T n,0 ∩T n,0 ̸= ∅.Then let us pull T 0 back by H◦ln .

It gives us the central interval T n+1,0 .29

Similarly we will construct a non-central interval T n+1,j(x) ≡T n+1,ji, x ∈ω(c) , as the pull-back ofappropriate T m,ji, m < n , corresponding to the first moment k when the orb(c) is separated from theorb(x) :H◦kn x ∈T m,ji,H◦kn c ̸∈T m,ji.Define hn+1|T n+1,j(x) = H◦(k+1)n.Central-low case. Let hnT n,0c ∈T n,0 and hnT n,0 ̸∋c .

Then let us consider the cascade of centralreturns until the first one which is low. It produces the intervals T n,jiby pulling T n,0iand T m,ji, (m

Now let us define a mapF :[m≤n, i̸=0T m,ji∪T n,j(n)0→T 0.For m < n set F| T m,ji= Hn .For m = n, i ̸= 0 , set F|T n,ji= H◦(j+1)n.Finally set F|T n,j(n)0= Hn .Now taking cut-offiterates F ◦mT n,j(n) we can construct the intervals T n+1,jiin the same way as inthe low case.Class ˜T . Observe that the above map F does not belong to class T : the image of the central intervalbelongs to int T 0 .

Such a situation always occurs in the end of a cascade of central returns. In order tohandle it we need to introduce a wider class of maps.Let us consider an interval T −1 whose interior contains finitely many disjoint closed intervals T 0i , withT 0 ≡T 00 containing c and symmetric with respect to it.

Let g : ∪T 0i →T −1 be a map with negativeSchwarzian derivative and a single critical point c satisfying the following properties:1) g diffeomorphically maps any non-central interval T 0i , i ̸= 0 onto T −1 ;2) g|T 0 is a composition of a quadratic map and a diffeomorphism onto T −1 with negative Schwarzianderivative;3) If g(T 0) ∩T 0i ̸= ∅and gc ̸∈T 1ithen g(T 0) ⊃T 1i(“Markov property”).4) gc ̸∈T 0 (non-central return).Let ˜T be the class of such maps.Observe that the first renormalization of a map of class ˜T belongs to class T . Moreover, if the scalingfactor µ = |T 0|/|T −1| is small then the scaling factor of the renormalized map is also small.

It followsthat the results of §3 are still valid if we start with a map of class ˜T : if µ < δ then the scaling factorsof renormalized maps go down to 0. Let us find the best δ satisfying this property.

Then given any smallσ > 0 , there is a map g ∈˜T such thatµ < δ(1 + σ). (5 −9)Let us start with such a map, and go through the general construction described above.

If this constructiondid not stop then we would have a map of class Twith arbitrary small initial scaling factor such that thescaling factors of the renormalized maps would not go to 0. Since this is impossible, our construction muststop.

Then we come up with a map H = HN of class ˜T (with T ′ = T N,j(N) as the central interval) suchthat HT ′ ⊃T ′ (high return). Since the scaling factors of H stay away from 0 (compare the above Remarkon the scaling factors), we conclude thatµ < |T ′||T 0|(1 + σ) < dist(Hc, c)|T 0|(1 + σ).

(5 −10)Push-forward. Let us consider now the interval B = T N,0 just constructed.

The map H = f ◦(l+1) isunimodal on B . Moreover, there is an interval A ⊃fB which is a monotone pull-back of T 0 by f l .Let p = (m, j) denote a point of the index set˜E = {(m, j) ∈E : m ≤Norm = N, j = 0}.30

We start with 0 ≡(0, 0) , and by p + 1 we mean the point of ˜E which lexicographically follows p . (So, weidentify the set ˜E ordered lexicographically with an interval of the set of integers p = 0, ..., P ).Let us consider now a sequence of intervals Ap = f ◦k(p)A ⊂T p−1p =, 0, ...P defined as the lastinterval of the orbit {f ◦mA}lm=0 visiting T p−1 , s(p) = k(p) −k(p + 1) .

Note that A0 = T 0 and k(0) = l ,while for p > 0, Ap ⊂T p−1∖T p . Moreover, f ◦s(p) diffeomorphically maps Ap+1 onto Ap , and the mapf ◦(s(p)−1)|fAp+1 has a Koebe space spread over T p−1 .

Also the Koebe space of f ◦k(P )A is spread overT P −1 .Finally let us mark in Ap the corresponding iterate ap = f ◦(s(p)+1) of the critical point. Then f ◦s(p)gives a diffeomorphism between corresponding marked intervals.

Note also that a0 = Hc .Distortion estimates. For 1 ≤p ≤P let the marked point ap divide Ap into intervals Lp and Gp withGp being closer to the critical point than Lp .

Set κp = |Gp|/|Lp| .Somewhat abusing notations let us denote by µn = |T n|/|T n−1| new scaling factors. Acting now as inthe above particular cases taking into account estimate (5-10) we obtain the following analogue of (5-5):κ0 ≤1µ1(1 + o(1))asσ →0.

(5 −11)Let us now estimate κp+1 through κp . We have (compare (5-6)):|Gp| ≤κp1 + κp|Ap| ≤κp1 + κp(1 −µp).

(5 −12)Let Rp be the component of T p−1∖Ap containing c . Then we can estimate the Poincar´e length P(Lp, Rp)as in (5-7):P(Lp, Rp) ≤log1 +2κp1 + µp≤log(1 + κpµp).

(5 −13)By the Schwarz lemma|fGp+1||fLp+1| ≤κpµp. (5 −14)Since non-linearity of f|Ap+1 is estimated by log(1/µp+1) ,κp+1 ≤κpµpµp+1.

(5 −15)Estimates (5-11) and (5-15) yieldκp ≤1(µ1...µP −1)2µP(1 + o(1)). (5 −16)Set now G′ = A∖fB .

Then using (5-14)-like estimate and (5-16) we conclude that|G′||fB| ≤1(µ1...µP )2 (1 + o(1)). (5 −17)Finally, we can actually improve this estimate as|G′||fB| ≤τ 21(µ1...µP )2 =τ |T 0||B|2(5 −18)with an absolute τ < 1 (see the argument preceding estimate (5-8)).Complex pull-backs.

Take now the disk D(T 0) , and pull it back by the branches of H . Then by (5-18)the central domain V0 based upon the interval B is compactly contained in D(T 0) .The non-centraldomains are compactly contained in D(T 0) , pairwise disjoint, and disjoint from V0 for the same reason asin the particular cases treated above.§6.

Polynomial-like maps.The following three pages, up to Lemma 6.3, present a self-contained exposition of a generalized versionof the Branner-Hubbard theory [BH], [B]. The reader can see the following differences.

We adjust the theory31

to a local setting of generalized polynomial-like maps which gives us a great flexibility in applications (see,e.g., [L2]) ( The original theory was all about cubic polynomials with one escaping critical point).Wetranslate it from the original tableau language to the language of pull-backs which nicely corresponds to theone-dimensional discussion of §2. Finally, we state the main rigidity result of the theory in the parameterplane as a lemma on qc conjugacy of polynomial-like maps with the same combinatorics (Lemma 6.3).

Adirect proof of this lemma was given by J.Kahn.After this preparation we complete the proof of Theorem B.Puzzle-pieces, kids and pull-backs. Let us remember that by a polynomial like map we mean an analyticbranched coveringf : ∪li=0Ui →Vwhere Ui and Vare topological disks, cl Ui ⊂V .

(The Douady-Hubbard polynomial-like maps [DH]correspond to l = 1 .) The setK(f) = {x : f ◦nx ∈∪Ui, n = 0, 1, ...}ofnon-escapingpointsiscalledthefilled-inJuliaset.ByAwedenoteaclassofpolynomial-like maps f with a single non-degenerate critical point c ∈U0 .

This class is a complex coun-terpart of the class T of one-dimensional maps.Figure 7Set V (n) = f −nV . The connected components V (n)kof V (n) are called puzzle-pieces of depthn .The puzzle piece of level n containing a point x will be also denoted by V (n)(x) .The puzzle-piecesV (n)0≡V (n)(c) containing the critical point are called critical.

The family of puzzle-pieces is Markov in thefollowing sense: for n > 0 V (n)(x) is mapped under f onto a puzzle piece V (n−1)(fx) . Moreover, thismap is a two-to-one branched covering if V (n)(x) is critical, and a conformal isomorphism otherwise.Let f ◦mx ∈V (n)k≡W .

Then we can pull the puzzle-piece W back along the orb m(x) and come upwith the puzzle-piece V (n+m)(x) ≡P . The pull-back is called univalent if the map f ◦m : P →W is.

It iscalled quadratic-like if P is critical and f ◦(m−1) : fP →W is univalent.A critical puzzle-piece P = V (n+m)0is called a kid of W = V (n)0if it is obtained by the quadratic-likepull back of W along the orb m(c) . If this corresponds to the first return of the critical point back to W ,then P is called the first kid of W .

Repeating this construction we can talk about grandkids of the n thgeneration.Let us say that f admits a quadratic-like renormalization if there is a critical piece W = V n0and itsfirst kid P = V n+m0such that the critical point does not escape P under iterates of f ◦m . In such a casef ◦m : P →W is a quadratic-like map with a connected Julia set.The critical point is called combinatorially recurrent if its orbit crosses all critical puzzle-pieces.Two kids lemma.

Assume that f ∈A does not admit a quadratic-like renormalization. Then each criticalpuzzle-piece has at least two kids.Proof.

Let us consider a critical puzzle-piece W = V (n)0and its first kid P = V (n+m)0. Since f does notadmit quadratic-like renormalizations, the critical point must escape P under some iterate f ◦mk .

Let k32

be the first escape moment; then f ◦kmc ∈W∖P . Since the critical point is combinatorially recurrent, wecan find the first return moment l of f ◦kmc back to W .

Then the puzzle-piece V (n−l)(f ◦kmc) ⊂W∖Pis obtained by the univalent pull-back of W along the orbl(f ◦kmc) . Pulling this piece further along theorb km(c) until it first hits the critical point, we will find the second kid.

⊔⊓Letusconsidernowmultiply-connecteddomainsA(n)(x)=V (n)(x)∖V (n+1) .The mod (A(n)(x) ) can be defined as the reciprocal of the extremal length of the family of (non-connected)curves separating the outer boundary component of A(n)(x) from all inner ones.The divergence property. Let f ∈A does not admit a quadratic-like renormalization.

Then for anyz ∈K(f)∞Xn=0mod(A(n)(z)) = ∞.Proof. Argue as Branner & Hubbard.

Let us concentrate on the principle case when the critical point isrecurrent, and z = c . All critical pieces are descendents of V = V (0)0, and can be graded by generations.By the previous lemma there are at least 2n grandkids in n th generation.Since mod( A(n)j) =mod(V )/2n for any such grandkid, the total sum of moduli over n th generation is at least mod (V ) .Hence the total sum of moduli over all descendents is ∞.

⊔⊓Corollary 6.1. A map f ∈A does not admit a quadratic-like renormalization if and only if the filled-inJulia set is Cantor.For this reason we also call maps which don’t admit quadratic-like renormalizations Cantor polynomial-like.

In this case K(f) certainly coincides with the Julia set.A set K ⊂C is called removable if given a neighborhood U ⊃K , any conformal/qc embeddingψ : U∖K →¯C allows conformal/qc continuation across K . The conformal and qc settings in this definitionare equivalent (by the Measurable Riemann Mapping Theorem).

The following important observation wasmade by Jeremy Kahn:Corollary 6.2. Assume that f ∈A does not admit a quadratic-like renormalization.

Then the Julia setK(f) is removable.Proof. This follows from the Modular Test on removability (see [SN], §1).

⊔⊓Talking about a conjugacy between two polynomial-like maps, we always mean local conjugacy inneighborhoods of their filled-in Julia sets.Lemma 6.3. Let f and g be two Cantor polynomial-like maps.

If they are topologically conjugate by ahomeomorphism h then they are qc conjugate by a qc map H which agrees with h on the Julia set.Proof. As in Lemma 2.10, let us set g = ˜f and mark the related objects by tilde.

Let h be a topologicalconjugacy. Select an N such that V (N) ⊂Dom(h) .

Let us consider an isotopy ht such that h0 = h , h1is smooth in a neighborhood of V (N)∖V (N+1) , h1 ◦f = g ◦h1 holds in a neighborhood of ∂V (N+1) , andht ≡h in a neighborhood of the filled-in Julia set. Since ht ≡h near K(f) , we can pull this isotopy backto V (N+1) in such a way that for the pull-back ht1 ≡h also holds near K(f) :ht1V (N+1)→˜V (N+1)f ↓↓˜fV (N)→˜V (N)htThen ht1 ≡ht in a neighborhood of ∂V (N+1) .

Hence we can continue ht1 to V (N) as ht .Now we can pull ht1 back in the same way, etc. We will obtain a sequence of isotopies htn such that(i) htn agree with h on K(f) ;(ii) htn agrees with htn−1 on V (N)∖V (N+n) ;(iii) ˜f ◦htn = htn−1 ◦f .33

(iv) h1n is smooth outside V (n) with a uniformly bounded qc dilatation (since the pull-backs by conformalmaps preserve the dilatation).Hence we can consider a family of maps Ht = lim htn where the limit is understood in a pointwise sense.Then Ht is an isotopy outside the Julia set, and H1|V (N)∖K(f) is a smooth qc map.Moreover, since the isotopy htn is concentrated in V (N+n) , it carries all puzzle-pieces V (N+n)(x) to thecorresponding pieces ˜V (N+n)(hx) . Hence H1 also carries V (N+n)(x) to ˜V (N+n)(hx) .

Since the diametersof these pieces shrink down to zero, we conclude that H1 is continuous across the Julia set, and agrees withh .By Corollary 6.2, H1 is actually qc. ⊔⊓In conclusion let us mention the following result which links local and global settings of the above theory(compare [DH]):Straightening Theorem.

Any polynomial-like map of class A is qc conjugate to a polynomial with onenon-escaping critical point. ⊔⊓R-symmetric case.

If the topological disks Ui and V are symmetric with respect to the real axis, and fpreserves the real axis, we call f the R -symmetric polynomial-like map.Lemma 6.4. If two R -symmetric polynomial-like maps f and g are topologically conjugate on the realline by then the conjugacy can be continued to the complex plane as well.Proof.

Let us continue the conjugating homeomorphism h to the complex plane in such a way that itrespects the dynamics on ∂V 1 . Now let us pull it back (as in lemmas 2.10 and 6.30), so that the pull-backshn : V →V agree with h on the real line.

Then there is a pointwise limit H = lim hn conjugating f andg . Since the puzzle-pieces shrink down to zero, H is a homeomorphism.

⊔⊓Lemma 6.5. A R-symmetric polynomial-like map f ∈A admits a quadratic-like renormalization if andonly if its restriction to the real line admits a unimodal renormalization.Proof.

Indeed, set J = V ∩R and J(n)i= V (n)i∩R provided the intersection is non-empty. Then J(n)iare exactly the intervals of the Markov family MJ (see §2).

Let V (n+m)0be the first kid of V (n)0. Thenthe property “orb( c ) does not escape V (n+m)0under iterated f ◦m ” is certainly equivalent to “orb( c )does not escape J(n+m)0under iterated f ◦m ”.

The former property means that f admits a quadratic-likerenormalization, while the latter one is equivalent to f|R admits a unimodal renormalization (see Lemma2.7). ⊔⊓Let us remember that Tmin denotes the class of maps f ∈Twith the recurrent c and the minimalcritical set ω(c) which does not admit unimodal renormalizations.

Putting together Theorem 2.12 and thelast three lemmas, we conclude:Theorem 6.6. Two R -symmetric polynomial-like maps of class Tmin with the same combinatorial type(that is ¯χf = ¯χg ) are qc conjugate.

⊔⊓Hence these maps are qs conjugate on the real line.The standard family. Given p, q ∈N and a spin function ǫ , let us consider a standard family of mapsof class T (p, q, ǫ)f t :p[k=−qIk →Jdefined after Lemma 2.7.

The quadratic central branch of f t depends on t while all non-central linearbranches are fixed. By Lemma 2.7 any admissible combinatorial type ¯χ can be realised in such a family.Since the lengths of the intervals Ik can be selected arbitrarily, we can construct a map f with a givencombinatorial type and arbitrarily small first scaling factor µ0 = |I0|/|J| .

By §3, the conclusion of TheoremB holds for such a map.Let us put the intervals Ik into J in such a way that they divide J into commensurable parts. Thenthe pull-back U0 of the Eucledian disk D(J) by the central branch has a bounded shape (regardless of34

the combinatorics and the lengths of Ik ). All non-central pull-backs Uk are true disks.

Hence if Ik aresufficiently small, f is polynomial-like.So, we have constructed a polynomial-like map f ∈Tmin with a given combinatorial type satisfying theconclusion of Theorem B: Poincar´e lengths of the gaps go up to ∞. But clearly this property is qs-invariant(since qs maps carry commensurable adjacent intervals to commensurable ones).

Now Theorem 6.5 yields:Lemma 6.7. Theorem B holds for any polynomial-like map f ∈Tmin .

⊔⊓Proof of Theorem B: concluding argument. Let us consider the subclass T ∗of maps f ∈T for whichthe conclusion of Theorem B is valid.

By the real argument of §3 and §4 this subclass includes all maps withthe non-minimal critical set, as well as maps with the minimal critical set of unbounded type.Let us supply T with a C1 -topology. (Observe that the classes T (p, q, ǫ) corresponding to the differentcombinatorics on the first level (see §2) stay far away even in C0 -topology.) By §3, T ∗is an open subspacein T .

Indeed, given an ǫ > 0 , the condition that there is an ǫ -small scaling factor µn specifies an openset of maps f . But by §3 this condition forces f to belong to T ∗, provided ǫ is sufficiently small.Let us take now a map f ∈Tmin of bounded type.

Assume that f ̸∈T ∗. Then by §4 we can select asequence of renormalized maps R◦n(k)fC1 -converging to a map g ∈Tmin of Epstein class.

By §5, thereis a polynomial-like renormalization h of g . By Lemma 6.7, h ∈T ∗, hence g ∈T ∗.Since T ∗is open, R◦n(k)f ∈T ∗for sufficiently large k .Hence f ∈T ∗as well, and this is aContradiction.The case of a quadratic polynomial: alternative argument.

This case can be treated in a morestraightforward manner skipping §§4,5. But then we need the Markov family of Yoccoz puzzle-pieces (see[H] or [M2]) and the renormalization construction of [L2].This construction goes as follows.

Let f be a non-tunable quadratic polynomial in the sense of [DH]. (A real quadratic polynomial is tunable if and only it admits a unimodal renormalization with period > 1 ).Take a critical puzzle-piece W = V (N)0such that V (N−1)0∖V (N)0is a non-degenerate annulus.

It satisfiesthe following “nice” property similar to (2-4):f ◦n(∂W) ∩clW = ∅, n = 1, 2, ...If ω(c) is a minimal Cantor set then we can pre-renormalize f on W in the same way as it was described in§2 for the real setting. Namely, consider the first return map to W and select only those pieces of its domainwhich intersect ω(c) .

We obtain a polynomial-like map g : ∪Ui →W . It does not admit a quadratic-likerenormalization since f is non-tunable.

Hence it is a Cantor polynomial-like map.Now if we start with a real quadratic polynomial f then Lemma 6.7 for g implies Theorem B for f .§7. Absence of attractors.Let f be a quasi-quadratic map topologically exact on T = [fc, f ◦2c] .

By [BL1-3] or by [GJ], [K],f|T has a unique measure-theoretic attractor A , that is, an invariant closed set such that ω(x) = A forLebesgue almost all x ∈T . Moreover, either A = T or A = ω(c) ∋c .

Our goal is to prove that the formercase holds. It is certainly true if c is not recurrent, or if ω(c) = T .Let us assume that c is recurrent and ω(c) ̸= T .

Then we can renormalize f on any nice interval I0 .Moreover, for I0 sufficiently short the domain of the pre-renormalized map g = g1 : ∪I1k →I0 is not dense,that isint(I0∖∪I1k) ̸= ∅. (7 −0)Since g does not admit unimodal renormalizations, the set K(g) of non-escaping points is nowhere dense(see the argument of Lemma 2.3).

Our goal is to prove that this set has zero Lebesgue measure.Combinatorics of the first return maps. Let I0 ⊃I1 ⊃I2 ⊃... be the sequence of the first kids of I0 .Let us construct inductively the return maps fn to In−1 .

Let Ln = ∪Lnibe the domain of definition ofthe fn where Lniare intervals with Ln0 as the central interval, Ln∗= ∪i̸=0Lni . Denote by Ln the familyof these intervals, and by Ln∗the family of non-central intervals.

Let Gn = int(In−1∖Ln) be the union ofgaps. All these sets are c -symmetric for n > 1 .To start induction observe that L1 = ∪I1k, f1 = g1 .

Moreover, G1 is non-empty by (7-0). Let usdenote by Gn∗the free semigroup generated by Ln∗.

For a word γ ∈Gn∗denote by the same letter γ the35

interval whose itinerary through Ln is given by γ . Let l be the length of γ .

Let us consider the subsetsγr and γe of this interval such that f ◦ln γn0 = In and f ◦ln γe = Gn , that is, the former subset goes to thecentral interval, while the latter one escapes through the gaps set Gn . Finally, let ∂Gn mean the set ofinfinite words in letters Ln∗which we identify with the corresponding fn -invariant Cantor set.

Since thisset does not contain the critical point, it is hyperbolic and has zero measure. Hence the sets ∪γe and ∪γrcover almost completely the set Ln∗.

LetEn = Gn [γ∈Gn∗γe,Rn = In [γ∈Gn∗γrbe the full sets of returning and escaping points. Because of the above remark, their union has full mea-sure in In−1 .

Moreover, we have a transition map Fn from Rn to the central interval In which mapsdiffeomorphically each interval γe onto In .Now, in order to construct the return map fn+1 of the next level, just consider the pull-back Ln+1 ofEn by the central branch fn|In , and define fn+1 = Fn ◦fn| Ln+1 .36

Density estimates.Lemma 7.1. Let n > 1 .

Then for any word γ ∈Gn∗|γe||γr| ≥12|Gn||In| .Proof. Let l be the length of the word γ .

Then f ◦lndiffeomorphically maps γ onto In−1 . By theMinimum Principle, there is a component L of In−1∖In such thatinfL Dg◦(−l)n≥supIn Dg◦(−l)n.Let H = Gn ∩L .

By the symmetry, |H| = (1/2)|Gn| . Hence|γe||γr| ≥|g◦(−l)nH||g◦(−l)nIn|≥|H||In| = 12|Gn||In| .⊔⊓Lemma 7.2.

Let U be any interval such that ∂U ⊂∂Ln ∪∂In−1 and U ∩In = ∅. Thendens(En|U)dens(Rn|U) ≥12|Gn||In| .Proof.

There is a family L of intervals γ ∈Gn∗such that we have the following coverings up to sets ofmeasure zero:(U ∩En)∖Gn =[γ∈Lγe (mod 0),U ∩Rn =[γ∈Lγr (mod 0).Apply now the previous lemma. ⊔⊓Let us state an elementary lemma about the quadratic map φ : x 7→(x −c)2 .Lemma 7.3.

Let X be a c -symmetric measurable set in a c -symmetric interval I . Thendens(X|I) ≥12dens(φX|φI).⊔⊓Let us remember the notation µn = |In|/|In−1| for the scaling factors.

Let δn = dens(Gn|In−1). If Uis any interval such that ∂U ⊂∂Ln ∪∂In−1 (but perhaps U ⊃In ) then Lemma 7.2 impliesdens(En|U)dens(Rn|U) ≥12δn1 −µnµn.

(7 −1)Pulling this back by fn|In = h ◦φ with the h of bounded distortion we obtain the following recurrentestimate:δn+1 ≥aδn1 −µnµn(7 −2)with an absolute a . It follows that δn+1 ≥2δn , provided µn ≤¯µ is sufficiently small.If µm > ¯µ is not too small then by the results of §3 we are in the tail of a cascade of central returns.Let m + q −1 be the first non-central return level of this cascade, q ≥1 .

Let H be the component ofIm+q−1∖Im+q containing the critical value fm+qc . Let us consider the intervals V = H ∩fm+qIm+q andU = V ∪Im+qkwhere fm+qc ∈Im+qk.

Then U satisfies the assumptions of Lemma 7.2. On the other hand,by §3, the Poincar´e length P(Im+qk|H) is very small.

We conclude thatdens(Gm+q|V ) ≥(1 −ǫ) dens(Gm+q|U)(7 −3)with a very small ǫ .Further, f q : U →f qU is a map of bounded distortion. Indeed, since |Im+q|/|Im| ≥τ ¯µ, τ > 0, staysaway from 0, fm has a bounded non-linearity on the interval Im∖Im+q .

Since the intervals f kU, k =37

0, ..., m, are pairwise disjoint, we have the bounded distortion of f q . Hence there is an absolute a > 0 suchthatdens(Gm+q|U) ≥a dens(Gm|f qU)(7 −4)Now Lemma 7.2 and estimates (7-3), (7-4) yielddens(Gm+q|V ) ≥aδm(7 −5)(as usually in analysis, an absolute constant a may have different values in different estimates).Now let us pass to the next level using Lemma 7.3.Let W ⊂Im+q be the pull back of Vbyfm+q|Im+q .

We conclude thatdens(Gm+q+1|W) ≥aδm(7 −6)with yet another a > 0 . On the other hand, by §3 |Im+q+1|/|W| is very small (exponentially small in termsof κ ).

Henceδm+q+1µm+q+1≥|Gm+q+1||Im+q+1| ≥dens(Gm+q+1|W)dens(Im+q+1|W) > Aδmwith a big A (for a big m ). Passing now to the next level using (7-2), we concludeTheorem 7.4.

The densities δn = dens(Gn|In−1) of gap sets stay away from 0, provided n is not in thetail of a long cascade of central returns or immediately after the cascade. Moreover, these densities grow atleast exponentially with κ(n) .Concluding argument.

Assume that the set K(g) of non-escaping points has positive measure. LetX = {x ∈K(g) : ω(x) ∋c} .

By [BL3], dens (X|In) = 1 (The argument: take a density point x ∈X andconsider the first moment l when glx ∈In . Then the corresponding pull-back T ∋x is mapped under glonto In with a bounded distortion.

)On the other hand, Theorem 7.4 says that dens (K(g)|In) →0 for an appropriate subsequence of levels.This contradiction completes the proof of theorem A.38

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cm+q+1m+qm+1 m.........g m+1c

chqRM

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