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On Post Critically Finite Polynomials

arXiv:math/9305207v1 [math.DS] 15 May 1993On Post Critically Finite PolynomialsPart One: Critical PortraitsAlfredo PoirierMath DepartmentSuny@StonyBrookStonyBrook, NY 11794-3651We extend the work of Bielefeld, Fisher and Hubbard on Critical Portraits (see [BFH]and [F]) to the case of arbitrary postcritically finite polynomials. This determines an effectiveclassification of postcritically finite polynomials as dynamical systems.This paper is the first in a series of two based on the author’s thesis (see [P]) whichdeals with the classification of postcritically finite polynomials.

In this first part we concludethe study of critical portraits initiated by Fisher (see [F]) and continued by Bielefeld, Fisherand Hubbard (see [BHF]). As an application of our results, we give in the second part ofthis series necessary and sufficient conditions for the realization of Hubbard Trees.1

Chapter IBasic Concepts and Main ResultsIn the first three sections of this introductory chapter we define the concept of criticallymarked polynomials and state their main combinatorial properties. Our definition extendsthe concept presented in [F] and [BFH] by including the possibility of periodic criticalpoints.

This definition differs slightly from that given in the above references in the strictlypreperiodic case, but our results are the same. This small modification will later be useful,because some proofs will be simplified.Our definition is supported by a number of examples given in Section 4.

We remarkhere that the ‘hierarchic selections’ in the construction, are essential only to the markingcorresponding to Fatou set critical cycles.Here they are needed in order to guaranteeuniqueness for the polynomial with specified critical portrait. (Compare Example 4.5, andsee the remark following Lemma II.2.4).

The inclusion of the ‘hierarchic selection’ for Juliaset critical points was made to uniformize notation and is not essential (compare [BFH]where all critical points are in the Julia set).1. Preliminaries.1.1.

Let P be a polynomial of degree d > 1 with Ω(P) the set of critical points. ForM ⊂C denote by O(M) = ∪∞n=0P ◦n(M) the orbit of M. If the orbit O(Ω(P)) of the criticalset is finite, we say that P is postcritically finite (PCF).

It follows that every critical pointof P is periodic or preperiodic. We call the orbit O(ω) of a periodic critical point ω (if any)a critical cycle.

In this postcritically finite case a criterion to decide when a preperiodic (orperiodic) point is in the Fatou set is as follows.A preperiodic point is in the Fatou set if and only if it eventually maps to a criticalcycle.If P is postcritically finite, then the Julia set J(P) and the filled in Julia set K(P) ofP are connected and locally connected (see [M] Theorem 17.5). As there are no wanderingdomains for the Fatou components of this polynomial P, each bounded Fatou componentcontains exactly one point z (called its center) which eventually maps to a critical point.If we map this component U(z) onto the unit disk by an uniformizing Riemann map φwith φ(z) = 0, we can talk about internal rays in U(z) defined as the preimages of radial2

segments under φ. Because we are in the locally connected case those internal rays can beextended up to the boundary.In the case of the basin of attraction of ∞, if the polynomial is monic and centered,the uniformizing Riemann map can be chosen tangent to the identity at ∞.

These rays arecalled external rays, and satisfy the condition P(Rθ) = Rdθ.In general, let ω 7→P(ω) 7→. .

. 7→P ◦n(ω) = ω be a critical cycle.

Then P ◦n : U(ω) 7→U(ω) is a degree D > 1 cover of itself (D is the product of the local degree of elements inthe orbit O(ω), and U(ω) the Fatou component with center ω). It follows then that theuniformizing Riemann map φω can be chosen so thatφω(z)D = φω(P ◦n(z)).In this case the Riemann map is known as the B¨ottcher coordinate (compare [M] Theorem6.7).

This coordinate is uniquely defined up to conjugation with a (D −1)th root of unity.In particular, it is easy to see that there are exactly D −1 ‘fixed’ internal rays, i.e, internalrays R satisfying P ◦n(R) = R. They correspond in the B¨ottcher coordinate to the segments{re2πkiD−1 : r ∈[0, 1), k = 0, . .

. , D −2}.What is important to note here, is that the same construction is valid for all elementsin the critical cycle.

Note that if we choose a coordinate φω in which the internal ray Rcorresponds to the real segment [0, 1), then we can choose in a unique way a coordinateφP (ω) (at P(ω)) for which P(R) corresponds to [0, 1). Furthermore in this caseφP (ω)(P(z)) = (φω(z))degωP ,where degωP is the local degree of P at ω (for more details see [DH1, Chapter 4, Proposition2.2]).1.2 Lemma.

If a critical point z belongs to a critical cycle of period n = nz, thenP ◦n|U(z) (which has degree say Dz > 1) has exactly Dz −1 different fixed points in theboundary ∂U of this component U(z) respect to this return map. Furthermore, all externalrays that land at such points have period exactly n.Proof.

The first part is well known. For the second, we consider near this periodicpoint segments of all the external rays which land there, together with the internal rayjoining this point to the center z.

The cyclic order of these analytic arcs must be preservedunder iteration. The result thus follows easily.#1.3 Supporting arguments.

Given a Fatou component U and a point p ∈∂U, thereare only a finite number of external rays Rθ1, . .

. , Rθk landing at p. These rays divide theplane in k regions.

We order the arguments of these rays in counterclockwise cyclic order{θ1, . .

. , θk}, so that U belongs to the region determined by Rθ1 and Rθ2 (θ1 = θ2 if there isa single ray landing at p).

The argument θ1 (respectively the ray Rθ1) is by definition the3

(left) supporting argument (respectively the (left) supporting ray) of the Fatou component U.In a completely analogous way we can define right supporting rays. Note that an argumentsupports at most one Fatou component (compare [DH1, Chapter VII.4]).

Furthermore, bydefinition, given a Fatou component U, for every boundary point p there is an external raylanding at p, and therefore a supporting ray for U.1.4 Extended Rays. Given an external ray Rθ supporting the Fatou component U(z)with center z, we extend Rθ by joining its landing point with z by an internal ray, and call thissetanextendedrayˆRθwithargument θ.1.5 Example.

Consider the postcritically finite polynomial Pc(z) = z2 + c (where thecritical value c ≈−0.12256117 + 0.74486177i satisfies c3 + 2c2 + c + 1 = 0). The rays withargument 1/7, 2/7, 4/7 all land at the same fixed point.

But R4/7 is the only ray landing atthis point, which supports the critical component. (Compare Figure 1.1.

)Figure 1.12. Construction of Critically Marked Polynomials.Given a postcritically finite polynomial P, we associate to every critical point a finitesubset of Q/Z and construct a critically marked polynomial (P, F = {F1, .

. .

, FnF }, J ={J1, . .

. , JnJ}).

Here Fk would be the set of arguments associated with the critical pointzFk in the Fatou set, and Jk would be the set associated with the critical point zJk in theJulia set. The number of elements in these finite sets would be equal to the local degree ofthe associated critical points.

We remark that given a polynomial its critical marking is notnecessarily unique. Also note that one of these two families will be empty if there are nocritical points in the Fatou or Julia set.

In the following definition we will always work withleft supporting rays. We remark that we could equally well work with the right analogue,4

but there must be the same choice throughout. Also, multiplication by d modulo 1 in R/Zwill be denoted by md.2.1 Construction of Fk.

First we consider the case in which a given Fatou criticalpointz=zFkisperiodic.Letz=zFk7→P(z)7→...7→P ◦n(z) = z be a critical cycle of period n and degree Dz > 1 (compare §1.1).Weconstruct the associated set Fℓfor every critical point zFℓin the cycle simultaneously. De-note by dz the local degree of P at z.

We pick any periodic point pz ∈∂U(z) of perioddividing n (which is not critical because it is periodic and belongs to the Julia set J(P))and consider the supporting ray Rθ for this component U(z) at pz. Note that this choicenaturally determines a periodic supporting ray for every Fatou component in the cycle.

Theperiod of this ray is exactly n (compare Lemma 1.2). Given this periodic supporting ray Rθ,we consider the dz supporting rays for this same component U(z) that are inverse imagesof P(Rθ) = Rmd(θ).

The set of arguments of these rays is defined to be Fk. Keeping inmind that a preferred periodic supporting ray has been already chosen, we repeat the sameconstruction for all critical points in this cycle.

Note that as the cycle has critical degreeDz, we can produce Dz −1 different possible choices for Fk. If Fk is the set associatedwith the periodic critical point zk, there is only one periodic argument in Fk (namely θas above), we call this angle the preferred supporting argument associated with zFk .

Notethat by definition, the period of zFk equals the period of the associated preferred periodicargument.Otherwise, if z = zFk of degree dz > 1, is a non periodic critical point in the Fatou setF(P), there exists a minimal n > 0 for which w = P ◦n(z) is critical. If w has associateda preferred supporting ray Rθ (at the beginning only periodic critical points do), thenP −n(Rθ) contains exactly dz rays which support this Fatou component U(z).

The set ofarguments of these rays is defined to be Fk. We pick any of those and call it the preferredsupporting argument associated with z.

We continue this process for all Fatou critical points.2.2 Construction of Jk. Given z = zJk (a critical point in J(P)) of degree dk > 1,we distinguish two cases.

If the forward orbit of z contains no other critical point, we havethat for some θ (usually non unique) Rθ lands at P(z). Now P −1(Rθ) consists of d differentrays, among them exactly dk land at z. Define Jk as the set of arguments of these rays,and choose a preferred ray.

Otherwise, z will map in n ≥1 iterations to a critical point,which we assume to have associated a preferred ray Rθ. In the nth inverse image P −n(Rθ)of this preferred ray, there are dk rays which land at z.

The set of arguments of these raysis defined to be Jk. Again we pick one of those to be preferred, and continue until everycritical point has an associated set.The critical marking itself gives information about how many iterates are needed for agiven critical point to become periodic.

For example we have the following lemma.2.3Lemma.LetγbeapreferredsupportingargumentinthesetFk(respectively in Jk).Then the multiple m◦nd (γ) (with n ≥1) is periodic but m◦n−1d(γ)is not if and only if zFk (respectively zJk ) falls in exactly n iterations into a periodic orbit.5

Proof. This clearly follows from the construction.#The importance of the above construction is stated in the following theorem.

The proofwill be given in Chapter III (compare also Theorem 3.9).2.4 Theorem. Every centered monic postcritically finite polynomial P has a criticalmarking (P, F, J ).

This marking determines the polynomial P in the following sense: if(P, F, J ) and (Q, F, J ) are critically marked polynomials, then P = Q. In other words,two monic centered post-critically finite polynomials with the same critical marking (F, J )must be equal.Remark.

Note that the construction of associated sets was done in several steps. Wefirst complete the choice for all critical cycles, and then proceed backwards.

In both theFatou and Julia set cases we will have to make decisions at several stages of the construction.Such decisions will affect the choice of the marking for all critical points found in thebackward orbit of these starting ones. Each time that this kind of construction is made, wewill informally say that it is a hierarchic selection.

We encourage the reader to take a lookat the examples in Section 4.6

3. The Combinatorics of Critically Marked Polynomials.In order to analyze which properties the families (F, J ) satisfy, it is convenient tointroduce some combinatorial notation.3.1 Definitions.

We say that a subset Λ ⊂R/Z is a (d-)preargument set if md(Λ)is a singleton. For technical reasons we will always assume that Λ contains at least twoelements.

If all elements of Λ are rational, we say that Λ is a rational preargument set. Itfollows by construction that whenever (P, F, J ) is a marked polynomial, all the sets Jk,and Fl are rational d-preargument sets.Consider now a family Λ = {Λ1, .

. .

, Λn} of finite subsets of the circle R/Z. The familyΛ determines the family union set Λ∪= S Λi.

We say that any λ ∈Λ∪is an element ofthe family Λ. Furthermore, we can say that it is a periodic or preperiodic element of thefamily if it is so with respect to md.

The set of all periodic elements in the family unionwill be denoted by Λ∪per.3.2 Hierarchic Families. We say that a family Λ is hierarchic if for any elementsin the family λ, λ′ ∈Λ∪, whenever m◦id (λ), m◦jd (λ′) ∈Λk for some i, j > 0 then m◦id (λ) =m◦jd (λ′).

(This is useful if we think of a dynamically preferred element in each Λk).3.3 Linkage Relations. We will say that two subsets T and T ′ of the circle R/Z areunlinked if they are contained in disjoint connected subsets of R/Z, or equivalently, if T ′is contained in just one connected component of the complement R/Z −T .

(In particularT and T ′ must be disjoint.) If we identify R/Z with the boundary of the unit disk, anequivalent condition would be that the convex closures of these sets are pairwise disjoint.If T and T ′ are not unlinked then either T ∩T ′ ̸= ∅or there are elements θ1, θ2 ∈T andθ′1, θ′2 ∈T ′ such that the cyclic order can be written θ1, θ′1, θ2, θ′2, θ1.

In this second casewe say that T and T ′ are linked. More generally a family Λ = {Λ1, .

. .

, Λn} is an unlinkedfamily if Λ1, . .

. , Λn are pairwise unlinked.

Alternatively, each Λi is completely contained ina component of R/Z −Λj for all j ̸= i.The preceding definition has its motivation in the description of the dynamics of externalrays for a polynomial map. Suppose the external rays Rθi, Rψi land at zi for i = 1, 2.

Ifz1 ̸= z2 then the sets {θ1, ψ1}, {θ2, ψ2} are unlinked, for otherwise the rays will cross eachother. The same argument applies if we consider rays supporting Fatou components.

But ifwe analyze linkage relations arising from rays supporting a Fatou component and rays thatland at some point, we may get minor problems. Anyway, it is easy to see that even in thiscase the associated sets of arguments will be ‘almost’ unlinked.

(Compare condition (c.2)and as well as Proposition 3.8 below. )3.4 Weak linkage relations.Consider two families F = {F1, .

. .

, Fn} and J ={J1, . .

. , Jm}; we say that J is weakly unlinked to F in the right if we can chose arbitrarily7

smallǫ>0sothatthefamily{F1, . .

. , Fn,J1 −ǫ, .

. .

, Jm −ǫ} is unlinked. (Here Λ −ǫ = {λ −ǫ (mod 1) : λ ∈Λ}.

)In particu-lar each family should be unlinked. Note that the definition allows empty families.

Tosimplify notation we will simply say that “F and J −are unlinked”.3.5 Formal Critical Portraits.Consider families F = {F1, . .

. , Fn} and J ={J1, .

. .

, Jm} of rational (d-)prearguments. We say that the pair (F, J ) is a degree d formalcritical portrait if the following conditions are satisfied.

(c.1) d −1 = P(#(Fk) −1) + P(#(Jl) −1)(c.2) F and J −are unlinked. (c.3) Each family is hierarchic.

(c.4) Given γ ∈F∪, there is an i > 0 such that m◦id (γ) ∈F∪per. (c.5) No θ ∈J ∪is periodic.This set of conditions represent the simplest conditions satisfied by the critical markingof a postcritically finite polynomial.

Condition (c.1) says that we have chosen the rightnumber of arguments. Condition (c.2) means that the rays and extended rays determinesectors which do not cross each other, and that F was constructed from arguments of leftsupporting rays.

This reflects our decision to chose the supporting arguments as the right-most possible argument of an external ray. Condition (c.3) reflects our choice of dynamicallypreferred rays.

Condition (c.4) indicates that arguments in F are related to Fatou criticalpoints. Condition (c.5) indicates that arguments in J are related to Julia set critical points.Unfortunately there are formal critical portraits which do not correspond to a postcriticallyfinite polynomial (compare Example II.2.8).

In order to state necessary and sufficient con-ditions we need to study the dynamically defined partitions of the unit circle determined bythese elements.3.6. Given two families F, J as above, we form a partition P = {L1, .

. .

, Ld} of theunit circle minus a finite number of points R/Z −F∪−J ∪, in the following way. Weconsider two points t, t′ ∈R/Z −F∪−J ∪.

By definition, t, t′ are unlink equivalent if theybelong to the same connected component of R/Z −Fi and R/Z −Jj, for all possible i, j.Let L1, . .

. , Ld be the resulting unlink equivalence classes with union R/Z −F∪−J ∪.

It iseasy to check that each Lp is a finite union of open intervals with total length 1/d.Each element Li ∈P of the partition is a finite union Li = ∪(xj, yj) of open connectedintervals.WealsodefinethesetsL+i=∪[xj, yj)andL−i=∪(xj, yj].ItiseasytoseethatbothP+={L+1 , . .

. , L+d }andP−= {L−1 , .

. .

, L−d } are partitions of the unit circle. As every θ ∈R/Z belongs to exactlyone set L+k , we define its right address A+(θ) = Lk.

In an analogous way we define the left ad-dress A−(θ) of θ. We associate to every argument θ ∈R/Z a right symbol sequence S+(θ) =(A+(θ), A+(md(θ)), .

. .) and a left symbol sequence S−(θ) = (A−(θ), A−(md(θ)), .

. .

). Notethat for all but a countable number of arguments θ ∈R/Z (namely the arguments present in8

the families and their iterated inverses), the left S−(θ) and the right S+(θ) symbol sequencescoincide. By S(θ) will be meant either (left or right) symbol sequence.3.7AdmissibleCriticalPortraits.LetF={F1, .

. .

, Fn}andJ = {J1, . .

. , Jm} be two families of rational (d-)prearguments.

We say that (F, J ) isa degree d admissible critical portrait if (F, J ) is a degree d formal critical portrait and thefollowing two extra conditions are satisfied. (c.6)Letγ∈F∪perandλ∈R/Z,thenλ=γifandonlyifS+(γ) = S+(λ).

(c.7) Let θ ∈Jl and θ′ ∈Jk. If for some i, S−(m◦id (θ)) = S−(θ′), then m◦id (θ) ∈Jk.3.8 Proposition.

If (P, F, J ) is a critically marked polynomial, then (F, J ) is anadmissible critical portrait.Condition (c.6) indicates that arguments in Fl must support Fatou components. Con-dition (c.7) indicates that different elements in the family J are associated with differentcritical points.

The proof of this proposition will be given in Section II.2.Now we can state the main result for critically marked polynomials as follows (the proofof this theorem will be given in Chapter III).3.9 Theorem. Let (F, J ) be a degree d admissible critical portrait.

Then there is aunique monic centered postcritically finite polynomial P, with critical marking (P, F, J ).Now we should ask if conditions (c.1)-(c.7) represent a finite amount of information tobe checked. This question is answered in a positive way by the following lemma.

The proofwould be given in Section II.1.3.10 Lemma. Suppose θ and θ′ have the same periodic left (or right) symbol sequence.Then θ and θ′ are both periodic and of the same period.In particular condition (c.6) can be replaced by condition (c.6)′:(c.6)′ Let γ ∈F∪per and let λ have the same period as γ, then λ = γ if and only ifS+(γ) = S+(λ).3.11.

The next question that we ask is what kind of information about the Julia setcan be gained by looking carefully into the combinatorics. For example, if can we determineif two rays land at the same point by only looking at their arguments.

In fact, left symbol9

sequences convey all the information necessary to effectively decide whether two rays landat the same point or not. This is done as follows.

Suppose Ji = {θ1, ..., θk} ∈J withcorresponding left symbol sequences S−(θ1), . .

. , S−(θk).

As we expect the rays with thosearguments to land at the same critical point, we declare them (i-)equivalent; i.e, we writeS−(θα) ≡i S−(θβ). Then we set θ ≈θ′ either if S−(θ) = S−(θ′) or there is an n ≥0 suchthat A−(m◦jd (θ)) = A−(m◦jd (θ′)) for all j < n and S−(m◦nd (θ)) ≡i S−(m◦nd (θ′)) for some i.This relation ≈is not necessarily an equivalence relation, because transitivity may fail.

Tomake this into an equivalence relation we say that θ ∼l θ′ if and only if there are argumentsλ0 = θ, λ1, . .

. , λm = θ′, such that λ0 ≈.

. .

≈λm. The importance of this equivalencerelation is shown by the following proposition.

The proof will be given in Chapter II.3.12 Proposition. Let (P, F, J ) be a critically marked polynomial.

Then Rθ and Rθ′land at the same point if and only if θ ∼l θ′.3.13 Corollary. The symbol sequence S−(θ) is a periodic sequence of period m if andonly if the landing point of the ray Rθ has period m.#We proceed now to give a very brief description of Chapters II and III which are devotedto critical portraits.

In Chapter II we will work in more detail the combinatorics of criticalportraits. We will also translate several of our results to the corresponding Julia set.

InChapter III we give the proof of the Realization Theorem for Critical Portraits. Appendix Adeals with the relevant part for our use of Thurston’s theory of postcritically finite rationalmaps.We state Thurston’s Theorem in a more general form.Namely, we include thepossibility of additional periodic or preperiodic orbits.

The proof given in [DH2] extends tothis formulation.We now give a brief comparison between our work and that by Bielefeld, Fisher andHubbard. Of course there is a big overlap in both expositions, and we have, when possible,referred to the original proofs.

In Chapter II, we analyze the combinatorics of the markingand the proofs follow the same lines as those in [BFH]. Chapter III is essentially differentand we have stated without proofs those results in [BFH] which apply to our case.

Themain point here is that as Levy cycles can not involve any ‘preperiodic element’ of thetopological polynomial, new preperiodic arguments should be introduced artificially. Wecall them special arguments.

Finally, we still have to prove that the recovered polynomialadmits the specified critical marking. Our method of proof is more delicate because newdifficulties are involved.Acknowledgement.

We will like to thank John Milnor for helpful discussions andsuggestions. Some of the arguments are in its final formulation thanks to him.

We willalso want to thank (among others) to Ben Bielefeld and John Hubbard for discussions atdifferent stages of the preparation of this work. Most of the figures were constructed using aprogram of Milnor.

Also, we want to thank the Geometry Center, University of Minnesotaand Universidad Cat´olica del Per´u for their material support.10

4. Examples.We will illustrate with examples the definitions of the previous sections.

We will try toisolate and illustrate all possible complications. Of course, the worst possible examples willinvolve several of these at the same time.4.1 The rabbit.

(See Figure 1.1.) Once again consider the degree two polynomialPc(z) = z2 + c with c ≈−0.12256117 + 0.74486177i.

The Fatou critical point z = 0 has aperiod 3 orbit under iteration. Therefore P ◦3 restricted to the critical component is a degree2 cover of itself.

It follows that the map P ◦3 has a unique fixed point in the boundary of thiscritical Fatou component. As noticed above, among the three rays R1/7, R2/7, R4/7 landingat this fixed point, only the ray R4/7 supports the critical component.

By the definitionof marking, we must look for the other ray that supports this component and maps toP(R4/7) = R1/7. This ray can only be R1/14.

Thus, we have constructed a marking for P.In this case F = {F1} and J = ∅, where F1 = {4/7, 1/14}.It is important to note that we were looking for a fixed point of P ◦3 restricted to theboundary of the critical Fatou component. Such a fixed point for P ◦3 turned out to be afixed point for P as well, but the rays landing there have period equal 3.4.2 The Ulam-von Neumann map.

We consider now the strictly preperiodic case.Let P(z) = z2 −2, and note that the orbit of the critical point z = 0 is 0 7→−2 7→2 7→2 . .

..Only the external ray R0 lands at z = 2, and therefore only the ray R1/2 lands at z = −2.Both R1/4, R3/4 land at the critical point z = 0, and map to R1/2 under P. In this case themarking is F = ∅and J = {J1}, where J1 = {1/4, 3/4}.4.3 Preperiodic case: two possible choices. (See Figure 1.2.) Consider the degreetwo polynomial Pc(z) = z2 + c where c ≈−1.5436891 is the only negative solution of theequation c3 + 2c2 + 2c + 2 = 0.Inthiscasethecriticalpointz=0hasorbit07→c7→c2 + c7→−(c2 + c) 7→−(c2 + c).

The rays R1/3, R2/3 both land at the fixed point z = −(c2 + c),and are interchanged by Pc.At z = c2 + c, the rays R1/6, R5/6 land.At the criti-cal value z = c, R5/12, R7/12.In this way, we can get two different markings F = ∅,J = {J1}, where J1 = {5/24, 17/24} corresponds to the choice of the critical ray R5/12,and J1 = {7/24, 19/24} to the choice of R7/12.In either case we can read from the marking that the critical point needs three iterationsto become periodic. The exact period however can not be read immediately from this data.

(Compare Corollary 3.13. )11

Figure 1.24.4 Non trivial critical cycle. (See Figure 1.3.) Consider the degree 3 polynomialP(z) = z3 −32z.

The critical points satisfy z2 = 1/2, and is easy to see that they areinterchanged by P (i.e, if a is a critical point then P(a) = −a). In each of the critical Fatoucomponents the map P ◦2 is a degree 4 (the product of the degrees of the cycle!) covering ofitself.

In this way, there must be in the boundary of each component 3 (= 4 −1) possiblechoices of periodic points. One of those fixed points (z = 0) belongs to the boundary ofboth components.

The rays landing at z = 0 are R1/4 and R3/4, and each one supportsexactly one of the Fatou critical components. The period 2 rays that support the ‘rightmost’component are R3/4, R7/8, R1/8 (their respective images R1/4, R5/8, R3/8 support the other).Therefore, the choice of a periodic supporting ray for one component, forces the choice ofits image for the other.Figure 1.3Thispolynomialhasexactlythreemarkings,alloftypeF={FA, FB},J = ∅.

The periodic supporting rays are listed on the left.Component AComponent BFAFBR3/4R1/4{3/4, 1/12}{1/4, 7/12}R7/8R5/8{7/8, 5/24}{5/8, 7/24}R1/8R3/8{1/8, 19/24}{3/8, 17/24}12

Thequestionisnow,whycanwenottakeFA={3/4, 1/12}andFB = {3/8, 17/24} as a marking? This is forbidden by the rules of §3 since 3/4 and 3/8 donot belong to the same cycle.

A good reason for this rule is given in the next example.Figure 1.44.5 Bad choice, wrong polynomial. (See Figure 1.4.) There is a polynomial withmarking F = {FA, FB}, J = ∅, where FA = {3/4, 1/12}, FB = {3/8, 17/24}.

But it is notthe one in Example 4.4.For the polynomial P(z) = z3 + az + b (where a = −0.75, b ≈0.661438i), the raysR3/4, R1/8, R1/4, R3/8, land at a fixed point which belongs to the boundary of the fourperiodic Fatou components. Those components are associated pairwise in cycles, so we havetwo disjoint degree 2 cycles.

Only R3/4 and R3/8 support critical components. It followseasily that this polynomial has a unique marking.4.6 Hierarchic choice.

(See Figure 1.5. )Consider now the polynomial P(z) =√2(z2 −1)2, with critical points z = 0, ±1.

The orbit of the critical points is ±1 7→0 7→√2 7→√2. At the fixed point z =√2 only the ray R0 lands.

At z = 0, R1/4 and R3/4. Atz = 1, R1/16, R3/16, R13/16, R15/16.

At z = −1, R5/16, R7/16, R9/16, R11/16. In this case themarking will not be unique and will depend in the choice of the preferred ray at z = 0.Figure 1.513

The marking will be of the form F = ∅, J = {Jz=0, Jz=1, Jz=−1}.Jz=0preferred rayJz=1Jz=−1at z = 0{1/4, 3/4}R1/4{1/16, 13/16}{5/16, 9/16}{1/4, 3/4}R3/4{3/16, 15/16}{7/16, 11/16}Figure 1.64.7 Badly mixed case. (See Figure 1.6.) Consider the degree 5 polynomial P(z) =c(z5 + 3z4 + 3z3 + z2), where c ≈4.3582708.

It has two Fatou critical components, one(on the right) fixed of degree 2, and one (on the left) preperiodic of degree 3 (absorbed bythe first in one iteration). The boundaries of these two Fatou components share a point,which happens to be critical.

The image of this Julia set critical point is the only fixedpoint lying in the boundary of the fixed Fatou critical component. Only the ray R0 lands atthis fixed point.

The rays R1/5, R4/5 are thus the only rays landing at the Julia set criticalpoint. Now, one of these rays (R4/5) supports the fixed Fatou component, while the othersupports the preperiodic one.

Also R0 must have two inverses supporting the fixed Fatoucomponent (R0, R4/5), and three supporting the preperiodic one (R1/5, R2/5, R3/5). Thus,the marking is F = {{0, 4/5}, {1/5, 2/5, 3/5}}, J = {{1/5, 4/5}}.

Note that in this casethere are arguments that belong to one family and to the other. Of course, if this happens,these arguments must be strictly preperiodic.By the moment we will take a closer look at condition (c.2) by analyzing this example.In this case, conditions (c.1), (c.3)-(c.5) are clearly satisfied.

To have a degree 5 formalcritical portrait, the three sets {0, 4/5}, {1/5, 2/5, 3/5}, {1/5 −ǫ, 4/5 −ǫ} must be unlinkedfor ǫ > 0 small; which is evidently true.4.8 Several critical cycles. (See Figure 1.3.

)Consider the degree 9 polynomialP ◦P where P is as in Example 4.4. The filled-in Julia set of this polynomial, as well as the14

external rays remain unchanged (with respect to P). In this case however, we have two fixedFatou components each of critical degree 4.

Each of them absorbs in one iteration anothercritical component. Now each cycle is independent of the other, and the choice of markingsare independent in the two fixed components.

Nevertheless, the choice of marking in thefixed Fatou components determines the marking of the critical components they absorb. Letus denote by A, B the fixed critical components and by A′, B′ the critical components theyabsorb.

The marking now is F = {FA, FA′, FB, FB′}, J = ∅.Component AFAFA′R3/4{3/4, 62/72, 6/72, 14/72}{30/72, 38/72}R7/8{7/8, 7/72, 15/72, 55/72}{31/72, 39/72}R1/8{1/8, 17/72, 57/72, 65/72}{41/72, 33/72}Component BFBFB′R1/4{1/4, 26/72, 42/72, 50/72}{66/72, 2/72}R5/8{5/8, 53/72, 21/72, 29/72}{31/72, 39/72}R3/8{3/8, 43/72, 51/72, 19/72}{5/72, 69/72}This implies that we have 9 possible markings. Note that the marking for the componentsA, B are independent, but they uniquely determine the marking for A′, B′.4.9 (See Figure 1.7.) In our final example we show the importance of working withtwo separate families F, J .

Consider the sets A = {0, 13}, B = { 59, 89}. The polynomialP(z) = z3 + Az + B (A = 2.25, B ≈−0.4330127i) has marking F = {A, B}, J = ∅, whilethe polynomial P(z) = z3+A′z+B′ (A′ ≈2.181104577, B′ ≈−0.3871686256i) has markingF = {A}, J = {B}.Figure 1.7.

Almost the same marking.15

Chapter IICritical Portraits.In this chapter we isolate the combinatorial properties of a critical portrait (F, J ) asdefined in Section I.3, and relate them to the dynamics of the respective critically markedpolynomial. Section 1 deals with the partition in the unit circle determined by this marking.We also prove here Lemma I.3.10.

Section 2 translates to the Julia set the language of Section1. As a consequence we prove that the critical marking defines an admissible critical portrait.In Section 3 we prove Proposition I.3.12 which gives the combinatorial criterion for decidingwhen two external rays land at the same point.

Section 4 characterizes the preimages ofmarked periodic rays landing at that same Fatou component from the combinatorial pointof view. Almost all the material in this chapter can be found in a weaker formulation in[BFH].

The essential novelty here is Section 4, which plays a central role in the proof of therealization Theorem for Critical Portraits.1. Partitions of the unit circle.In this section we fix a formal critical portrait (F, J ), and study some dynamicalproperties of the partition determined by these families.Given a formal critical portrait (F, J ), we defined in Chapter I the partitions P ={L1, .

. .

, Ld} and P± = {L±1 , . .

. , L±d }.

The first partition omits the arguments in F∪∪J ∪;while the other two cover the whole circle R/Z. We also know that each Lp (L±p ) is a finiteunion of open (semiopen) intervals with total length 1/d (compare Section I.3.6).

From thedynamical point of view we can say even more.1.1 Lemma. Each Lp is mapped bijectively by md onto the complement of a finiteset.

Each L±p is mapped bijectively by md onto the whole unit circle. Furthermore thesecorrespondences preserve the circular order.Proof.

The proof is straightforward and is left to the reader.#Before the next corollary, we recall briefly the standard language for manipulation ofsymbol sequences. Let S = (S0, S1, .

. .

), where Si ∈P. The shift of S is the sequence16

σ(S) = (S1, S2, . .

. ).

(Formally σ is a map from the space of symbol sequences to itself. )The ith projection πi is the map from symbol sequences to the partition space P defined byπi(S) = Si.

The proof of the following corollary is an easy induction using Lemma 1.1 andis left to the reader.1.2 Corollary. Suppose m◦nd (θ) = m◦nd (θ′) and πj(S+(θ)) = πj(S+(θ′)) for all j < n,then θ = θ′.

(The same is true if we consider left symbol sequences instead.)#Warning. Corollary 1.2 is not necessarily true if we compare left with right symbolsequences.

From S+(θ) = S−(θ′) and md(θ) = md(θ′), we can not infer θ = θ′. For example,in the Ulam-von Neumann map (compare Example I.4.2), S+(1/4) = S−(3/4), and botharguments become equal after doubling.As our partitions are well behaved under iteration, it is natural to introduce dynamicallydefined refinements.

The fact that these refinements are also unlinked allow us derive somebasic properties of symbol sequences.1.3Definition.ForS0, S1, . .

.∈P,setUS0,...,Sn={θ∈R/Z:m◦id θ ∈Si, i = 0, ..., n}. The Lebesgue measure of this set is 1/dn+1 as can be easily verifiedby induction.

Also set US0,S1,... = T∞n=0 cl(US0,...,Sn). This last set being a nested intersec-tion of non empty compact sets, is non empty.

It is easy to see that if S(θ) = (S0, S1, S2, . .

. )then θ ∈US0,S1,S2,....

It follows that given S0, S1, . .

. ∈P, there exists an argument whichhas either left or right symbol sequence (S0, S1, S2, .

. .

).1.4 Lemma. For each n ≥0 the family {US0,...,Sn} is unlinked.Proof.

This follows by construction and Lemma 1.1.#1.5 Lemma. There are only a finite number of arguments which admit a given symbolsequence.Proof.

Consider the full orbit of both families Λ = O(F∪) ∪O(J ∪). It is enough toprove that the number of connected components of US0,S1,...,Sn −Λ is bounded by a numberwhich depends only on (F, J ).

We claim that the cardinality N = #(Λ) of Λ is the boundwe are looking for. We prove this by induction.

For n = 0 this is clear. Now supposeUS1,S2,...,Sn −Λ = ∪ki=1I1, where each Iα is connected and k ≤N.

By construction every setS0∩m−1d (Iα) is completely contained in a component of R/Z−Λ and therefore is connected.The result follows.#1.6 Lemma. Suppose θ, θ′ have the same periodic left (or right) symbol sequence.

Thenθ, θ′ are periodic and have the same period.17

Proof. First note that θ can not be strictly preperiodic.

For otherwise, eventuallyit becomes periodic, and such periodic argument would have at least two different inverseswith the same symbol sequence, in contradiction with Corollary 1.2. If θ, θ′ are periodicof different period, we assume without loss of generality that θ is fixed, but θ′ is not.

Inthis case, we have at least three points with the same symbol sequence, for which the cyclicorder is not preserved under iteration, but this is a contradiction to Lemma 1.1. Finally, θcan not be irrational because of Lemma 1.5.#1.7 Remark.

We conclude this section with a trivial remark that will be used laterseveral times. If we take θ, θ′ ∈Jk and λ such that A−(λ) = A−(θ), then by definitionλ ∈(θ′, θ].

Analogously, if θ, θ′ ∈Fk and λ is such that A+(λ) = A+(θ), then λ ∈[θ, θ′). (There is nothing special about J or F in this formulation; but this is the way in whichthese statements will be used.

)18

2. The induced partitions in the dynamical plane.In this section we introduce the induced partition of the Julia set with respect to thegiven critical marking.

The main result is that this partition is Markov. As a consequenceof this, we establish that the critical marking of a postcritically finite polynomial is in factan admissible critical portrait, establishing in this way Proposition I.3.8.Let (P, F, J ) be critically marked.

In analogy with the way we constructed a partitionP of the unit circle where only the arguments in F∪∪J ∪were omited, we will constructa partition of the dynamical plane offthe rays with argument in J ∪and extended rayswith argument in F∪. To simplify this construction we introduce some notation.

For a setΛ ⊂R/Z we denote by R(Λ) the set of all external rays with argument in Λ and theirlanding points. Also, whenever Λ ⊂R/Z is a set of arguments each of them supporting aFatou component, we denote by E(Λ) the set of all extended rays with argument in Λ andthe respective centers of Fatou components.Figure 2.1 The critically marked polynomial P(z) = z3 + 1.5z with critical portrait (F ={{0, 1/3}, {1/2, 5/6}}, J = ∅) determines a partition of the dynamical plane.

However theelements of this partition are not necessarily connected open sets. Note that 0 and 1/2 sharethe same left symbol sequence in the circle, while the rays R0 and R1/2 land at the samepoint in the dynamical plane.Definition.

We say that two points z1, z2 in C −R(J ∪) −E(F∪) are “unlink equiva-lent”, if they belong to the same connected component of C −R(Ji) and of C −E(Fl) forall possible choices of Ji and Fl in the marking.Looking at the circle at infinity we immediately derive some properties. First, it iseasy to see that there are exactly d (=deg P) equivalence classes.Next, we have thateither an external ray is completely contained in an equivalence class, or is disjoint from it.Furthermore, we have that two rays Rθ and Rθ′ belong to the same equivalence class if and19

only if their arguments θ and θ′ belong to the same element S ∈P. Thus, these equivalenceclasses are in canonical correspondence with the elements of the partition P. For S ∈P wedenote by US the corresponding equivalence class in the dynamical plane.

Each equivalenceclass is by definition a finite union of unbounded open sets. Note that if two argumentsbelong to the same connected component of some S ∈P, then the respective rays will becontained within the same connected open region in the dynamical plane.2.1 Lemma.

Each region US is mapped bijectively by P into the complement of a finitenumber of rays and extended rays.#2.2 Lemma. The closure cl(US) and its restriction to the Julia set JS = J(P)∩cl(US)are connected.#Both proofs are somehow trivial and are left to the reader (compare also the proofs ofLemma 2.3 and Corollary 2.4).We can go a step beyond, and take the regions determined by the n-fold inverse imagesof those rays and extended rays.

Or alternatively we can dynamically define sets US0,...,Snin analogy with §1.3. The analogy between this and the definition given in §1.3, is clear: bydefinition, Rθ ⊂US0,...,Sn if and only if θ ∈US0,...,Sn.

Even if the sets US0,...,Sn are usuallydisconnected we have that their closures are not.2.3 Lemma. Let γ : [0, 1] →C be an arc which crosses neither a ray with argumentin O(md(J ∪)) nor an extended ray with argument in O(md(F∪)).

Suppose further that theimage of γ is disjoint from the forward orbit of all Fatou critical points. If γ contains aninterior point disjoint from these rays and extended rays, then γ can be lifted in a uniqueway within any cl(US), for all S ∈P.Proof.

Pick an S ∈P and start the lifting of γ at an image point not in the aboverays or extended rays. Note that the hypothesis guarantees that the lifting can be chosenin such way that it never gets into any region US′ other than US.

Uniqueness follows fromLemma 2.1.#2.4 Corollary. The closure cl(US0,...,Sn) and its restriction to the Julia set JS0,...,Sn =J(P) ∩cl(US0,...,Sn) are connected.Proof.

Note that if we cut open the plane along all extended rays with argumentin O(md(F∪)) and remove the forward orbit of all Fatou critical points, we are left witha connected set. In fact, given a Fatou component U, there is at most one argument inO(md(F∪)) which supports U.

This follows by construction of critical marking using thehierarchic selection. (This is the only place where the hierarchic selection is essentially used20

in this work!) Therefore we can join any two points in the Julia set with a path satisfyingthe hypothesis of Lemma 2.3.

The result now follows by induction on n.#Remark. That JS0,...,Sn is connected depends upon the fact that the definition ofcritical marking follows a hierarchic selection.

Without hierarchic selection for extendedsupporting rays, the statement above is definitely not true.At the end, we are mostly interested in the effect of this partition in the Julia set. Weset JS0,S1,... = T∞n=0 JS0,...,Sn Note that because J(P) is locally connected, it follows easilythat the external ray Rθ lands somewhere in the set JS+(θ) ∩JS−(θ).

Therefore we shouldask if JS(θ) consists of exactly one point.2.5 Lemma. For any sequence (S0, S1, .

. .) the set JS0,S1,... contains exactly one point.Proof (Compare with [GM, Lemma 4.2]) We will make use of the Thurston orbifoldmetric associated with P.Let MP be the surface with boundary, equal to the disjointunion of all ˜US defined as cl(US) cut open along all marked rays, extended rays and theirforward images, and with the orbit of the Fatou critical points removed.

Define the distanceρ(z, z′) between two points of MP to be the infimum of the lengths with respect to theorbifold metric of smooth paths joining z to z′ within MP (or ∞if they belong to differentcomponents). If z and z′ belong to the same subset JS0,S1 ⊂J(P), then any path from P(z)to P(z′) within ˜US1 can be lifted back uniquely to a path from z to z′ within ˜US0 (compareLemma 2.3).

Since the orbifold metric is locally strictly expanding, a compactness argumentshows thatρ(P(z), P(z′)) ≥cρ(z, z′)for some constant c > 1, independent of Si for this P. Therefore, the inverse mapP −1S0 : JS 7→JS0,Scontracts lengths by at least 1/c. Hence the iterated inverse images P −1S0 ◦.

. .

◦P −1Sn (JSn+1)have diameter less than some constant divided by 1/cn. Taking the limit as n →∞, weobtain the required unique point.#2.6 Corollary.

For any sequence (S0, S1, . .

.) we have P(JS0,S1,...) = JS1,S2,....Proof.

For some θ, either its left or right symbol sequence S(θ) equals (S0, S1, . .

. ).

Asthe ray Rθ lands at the unique point contained in JS0,S1,..., the result follows.#2.7 Corollary. If (S0, S1, .

. .) is a periodic sequence of period m, then the unique pointin JS0,S1,... is periodic of period dividing m.Proof.

This follows from Lemma 2.5 and Corollary 2.6. In fact, the period is m butthis is not a priori obvious, this will follow from Proposition 3.6.#21

2.8 A formal critical portrait not coming from a polynomial. Consider thedegree 4 formal critical portraitJ = {{ 360, 1860}, {1960, 3460}, { 160, 4660}},which does not came from the marking of a polynomial.

(Compare condition (c.7) in §I.3.7and Corollary 2.9, here S−(19/60) = S−(46/60)).Figure 2.2. Julia set of P(z) = z4 + Az2 + Bz + C with the rays160,360, 1860, 1960, 3160, 3460, 4660,4960 shown.

HereA ≈0.38437710 −0.56951210iB ≈0.30830201 + 0.03253718iC ≈0.49119643 + 0.93292127iIf there is a polynomial P of degree 4 which realizes this critical portrait, there shouldbe critical points ω1 ̸= ω2 associated with { 1960, 3460} and { 160, 4660} respectively.But asS−(19/60) = S−(46/60), then Lemma 2.5 tell us ω1 = ω2. Thus, the critical points as-sociated with { 1960, 3460}, { 160, 4660} must be actually the same.

Therefore we do not have threedegree 2 critical points, but one of degree 3 and the other of degree 2. In this case, therays R4/60, and R16/60 land at the same fixed point.

This fixed point has exactly one otherpreimage, the degree 3 critical point. At this critical point the rays R19/60, R34/60, R49/60,R1/60, R31/60, and R46/60 land.Therefore, the actual polynomial must have as criticalmarking either of the following,J = {{ 360, 1860}, {1960, 3460, 4960}},orJ = {{ 360, 1860}, { 160, 3160, 4660}}.2.9 Corollary.

Let (P, F, J ) be a critically marked polynomial. Suppose θ ∈Jk andθ′ ∈Jl.

If S−(m◦id (θ)) = S−(θ′) for some i ≥0, then m◦id (θ) ∈Jl.22

Proof. It follows from Lemma 2.5 that the rays with argument m◦id (θ) and θ′ landat the same critical point.

The result then follows from the hierarchic selection of rays. (Compare §I.2.

)#2.10 Corollary.Let γ ∈F∪per, and λ ∈R/Z, then λ = γ if and only ifS+(γ) = S+(λ).Proof. Suppose Fk = {γ = γ1, .

. .

, γn}, where the arguments γ1, . .

. , γn are in coun-terclockwise cyclic order.

Suppose λ ̸= γ but S+(γ) = S+(λ). By Lemma 2.5 the raysRγ, Rλ land at the same point.

As λ is periodic by Lemma 1.6, it follows that λ ̸∈Fk.But then, by definition of the right address A+(λ) of λ, it follows that the cyclic order isγ1, λ, γ2, . .

. , γn (compare Remark 1.7).

By definition of supporting argument (see §I.1.3),the corresponding Fatou component must be in the sector determined by Rγ1, Rλ (in thecounterclockwise sense). But this is a contradiction with the fact that Rγ2, .

. .

, Rγn alsosupport this component.#The following now follows from Corollaries 2.9 and 2.10.2.11 Proposition. If (P, F, J ) is a marked polynomial, then the pair (F, J ) is anadmissible critical portrait.#23

3. Which rays land at the same point?We would like to have a combinatorial criterion to decide when two rays land at thesame point.

Two arguments θ, θ′ in the same Jk do not have equal (left or right) symbolsequences. Nevertheless, the external rays Rθ, Rθ′ both land at the same critical point.

Ingeneral, all exceptions are a consequence of this fact. Furthermore, all the information weneed is already contained in left symbol sequences.3.1 The landing equivalence (∼l).

We recall briefly the definition of the “landingequivalence” ∼l between angles, introduced in Chapter I (compare §I.3.11). Let (F, J ) be anadmissible critical portrait.

For θα, θβ ∈Ji ∈J we set S−(θα) ≡i S−(θβ). Then we writeθ ≈θ′ if either S−(θ) = S−(θ′) or there is an n ≥0 such that πj(S−(θ)) = πj(S−(θ′)) forall j < n and σn(S−(θ)) ≡i σn(S−(θ′)) for some i.

Finally we make this into an equivalencerelation by letting θ ∼l θ′ if and only if there are arguments λ0 = θ, λ1, . .

. , λm = θ′, suchthat λ0 ≈.

. .

≈λm. Note that condition (c.7) together with (c.3) guarantee that wheneverθi ∈Fi (i = 0, 1); then θ0 ∼l θ1 if and only if F1 = F2.If the family J is empty, two arguments are equivalent if and only if their left symbolsequences coincide.

As S−(θ) is strictly preperiodic for every argument θ in the family unionJ ∪, two periodic or irrational arguments θ, θ′ are ∼l equivalent if and only if S−(θ) = S−(θ′).Of course, a preperiodic argument would never be equivalent to a non preperiodic one.By definition, if θ ∼l θ′ there is an m ≥0 such that σm(S−(θ)) = σm(S−(θ′)). Alsonote that whenever θ ≈θ′ then also md(θ) ≈md(θ′).

Therefore the following lemma istrivial.3.2 Lemma. If θ ∼l θ′ then md(θ) ∼l md(θ′).#Now let (P, F, J ) be a critically marked polynomial.We will show now that the∼l equivalence classes defined from the associated admissible critical portrait effectivelycharacterize the arguments of rays landing at a common point.3.3 Lemma.

Suppose Rθ, Rθ′ both land at the same point z. If z is non critical thenA−(θ) = A−(θ′).Proof.

If z is not the landing point of a ray with argument in F∪, then it is in theinterior of some region US. Otherwise, let Rθ1, .

. .

, Rθk be all rays with argument in F∪landing at z. Around z we consider locally segments of these rays together with internal raysjoining this point z to the center of the k associated Fatou components.

This configurationdivides a neighborhood of z into 2k consecutive regions. As every other region is containedin US where S = A−(θ1) = .

. .

= A−(θk), the result follows.#24

3.4 Lemma. Suppose Rθ lands at a critical point ω, then A−(θ) = A−(θ′) for someθ′ ∈Jω.Proof.

The external ray Rθ is contained within some UA−(θ′).#3.5 Corollary. Suppose θ, θ′ are such that A−(θ) = A−(θ′).

Then Rθ, Rθ′ land at thesame point if and only if Rmd(θ), Rmd(θ′) land at the same point.#3.6 Proposition. Let (P, F, J ) be a marked polynomial.

Then Rθ and Rθ′ land at thesame point if and only if θ ∼l θ′.Proof. First suppose that θ ∼l θ′.

If S−(θ) = S−(θ′) then the rays Rθ, Rθ′ land atthe same point by Lemma 2.5. Otherwise, it is enough to assume θ ≈θ′.

In this way, forsome n ≥0, σn(S−(θ)) ≡i σn(S−(θ′)) and πj(S−(θ)) = πj(S−(θ′)) for j < n. By definitionthere are arguments in Ji with symbol sequences σn(S−(θ)) and σn(S−(θ′)). As the rayswith these arguments land at the same critical point ωi, the rays Rm◦nd (θ) and Rm◦nd (θ′) alsoland at ωi.

The result follows now from Corollary 3.5.Conversely, suppose Rθ and Rθ′ land at the same point z. There is a minimal m ≥0such that P ◦m(z) neither is critical nor contains a critical point in its forward orbit.

We willprove by induction in m that θ ∼l θ′. Let P ◦n(z) be non critical for all n ≥0 (this is thecase m = 0).

For all n ≥0, Rd◦n(θ), Rd◦n(θ′) will be rays landing at the same non criticalpoint. In this case the result follows from Lemma 3.3.

Now, let md(θ) ∼l md(θ′) (this is theinductive hypothesis). If z is not a critical point we use again Lemma 3.3; if z is a criticalpoint we use Lemma 3.4.

In either case we deduce that θ ∼l θ′.#3.7 Corollary. If (S0, S1, .

. .) is a periodic sequence of period m, then the unique pointin JS0,S1,... has period m.#4.

Which rays support the same Fatou component?In general it is impossible to give a combinatorial description of when two argumentssupport the same Fatou component. This because the closure of two Fatou componentsmay share a periodic point which is not the landing point of a marked ray.

In this case,the arguments of all rays landing at such point will have the same left and right symbolsequences, and thus they are undistinguishable from the combinatoric point of view. How-ever, for some cases we will study which rays support some given periodic Fatou component.We will only consider rays for which some forward image belongs to the periodic part of thefamily union F∪per.

The importance of the combinatorial construction below will becomeclear in the next chapter. In the meanwhile we can tell the reader that in order to applythe theory of “Levy Cycles” (compare Appendix A), we should artificially introduce some25

preperiodic arguments for every periodic critical point. These preperiodic arguments arewhat we call in this section “special arguments”.As motivation for the combinatorial construction to follow, we consider a criticallymarked polynomial (P, F, J ).Let γ ∈O(F∪per), be of period k.Suppose also thatm◦nkd(λ) = γ.4.1 Lemma.

With the above hypothesis, Rλ supports the same Fatou component asRγ if and only if, for each i ≥0 eitheri) πi(S+γ) = πi(S+λ), orii) m◦id (γ) belongs to some Fα and A+(m◦id (λ)) = A+(γ′) for some γ′ ∈Fα.Proof. The proof is straightforward and is left to the reader.#This motivates the following definition.4.2 Special arguments.Let (F, J ) be an admissible critical portrait.To everyγ ∈O(F∪per) we associate a (periodic) sequence of sets T (γ, i) as follows.

First we defineT (γ, 0):T (γ, 0) ={A+(γ′) : γ′ ∈Fα}if γ ∈Fα for some α;{A+(γ)}otherwise.In the general case set T (γ, j) = T (m◦jd (γ), 0).Definition.Let γ ∈O(F∪per) be of period k = k(γ).We say that λ is a specialargument for γ, if there is an n ≥0 such that πi(S+(λ)) ∈T (γ, i) for all i < nk andm◦nkd(λ) = γ. In case both θ and θ′ are special arguments for γ ∈O(F∪per) we write θ ∼γ θ′.The following establishes an equivalence relation between ‘special arguments’.4.3 Lemma.

If λ is a special argument for both γ, γ′, then γ = γ′.Proof. Let n be a multiple of k(γ)k(γ′) big enough, then S+(γ) = σn(S+(λ)) = S+(γ′)and the result follows from condition (c.6) in the definition of admissible critical portraitand Corollary 1.2.#4.4 Remark.

If θ ∼γ θ′ and S+(θ) = S+(θ′), it follows from the definition of ∼γ,condition (c.6) and Corollary 1.2 that θ = θ′.26

These relations between special arguments are compatible with md in the followingsense.4.5 Lemma. If λ1 ∼γ λ2 then md(λ1) ∼md(γ) md(λ2).Proof.

For some high iterate γ = m◦kd (λ1) = m◦kd (λ2). Thus md(γ) = m◦kd (md(λ1)) =m◦kd (md(λ2)) and the result follows from the definition of ∼md(γ).#The following proposition, is a technical result needed in the proof of the main theorem(Theorem I.3.9).

Its meaning when translated to the context of PCF polynomials, is thatinverse images of a (marked) periodic ray supporting that same Fatou component, can befound very close to the starting periodic ray (this is obvious in the context of dynamics,because we are in the subhyperbolic case).4.6Proposition.Let(F, J )beanadmissiblecriticalportrait.If γ ∈F∪per then there exist arbitrary small ǫ > 0 such that γ + ǫ ∼γ γ.Proof. Let Sγ = (A+(γ), .

. .

, A+(m◦k−1d(γ))) and take any W ∈T 0γ × . .

. × T k−1γdifferent from Sγ.

We form a sequence γn ∼γ γ, where S+(γn) = SnγW ¯Sγ. Take a convergentsubsequence to λ.

As S+(λ) = ¯Sγ = S+(γ) it follows by condition (c.6) that λ = γ. Now,for ǫ > 0 small enough, γn can not be of the form γ −ǫ by Remark 1.7, therefore it must beof the form γ + ǫ.#In the language of special arguments Lemma 4.1 reads.4.7Proposition.Let(P, F, J )beamarkedpolynomial.Ifθisaspecial argument for γ ∈F∪per then Rθ and Rγ support the same Fatou component.#27

Chapter IIIRealizing Critical PortraitsIn this Chapter we give the proof of the Realization Theorem for Critical Portraits. InSection 1 we prove that the combinatorial data is ‘compatible’ in the sense that it allows usto construct a Topological Polynomial.

The actual construction is carried out in Section 2,where we also indicate (following [BFH]) that it is essentially unique. In Section 3 we provethat every admissible critical portrait has associated a unique (up to affine conjugation)polynomial which is Thurston equivalent to the topological polynomial so far constructed.In Section 4 we show that the isotopies between the ‘actual’ and ‘topological’ polynomials canbe chosen fixed not only relative to certain ‘marked’ points, but also relative to the wholeboundary when suitably chosen neighborhoods of Fatou points are deleted.

In Section 5we complete the proof of the Theorem by assigning the expected critical marking to theassociated polynomial.1. Combinatorial Information of Admissible Critical Portraits.In this Section we analyze the linkage relations that arise when we consider the full orbitof the families and special arguments together.

The main result is summarized in Proposition1.2 and is used in Section 2. This fact is easy to believe but its proof is extremely technical.1.1 Consider an admissible critical portrait (F, J ).

The orbit set O(F∪) can be parti-tioned in a natural way as F ∪{{γ} : γ ∈O(F∪) −F∪}. In the context of dynamics, twoelements in the orbit O(F∪) belong to the same element of this partition if and only if theysupport the same Fatou component (compare Proposition II.4.7).

If in addition we considera finite invariant set of special arguments Γ (i.e, satisfying md(Γ) ⊂Γ∪F∪), we can includean element λ ∈Γ in that same class as γ, whenever λ ∼γ γ. In this way, we construct afamily F∗= {F∗1 , .

. .

, F∗n} which is a partition of O(F∪) ∪Γ.Next, we partition the set O(F∪) ∪O(J ∪) ∪Γ ∪{0} into ∼l equivalence classes toform the family J ∗= {J ∗1 , . .

. , J ∗m}.

In the PCF context we are grouping all those rayswe expect to land at the same point (compare Proposition II.3.6). Here we are adding theargument θ = 0 to simplify things later.

This will reflect the choice of R0 as a preferredfixed ‘internal’ ray in the basin of attraction of ∞. (Compare Example 3.7.

)In the way the pair (F∗, J ∗) was constructed, it is clear that if we think in terms ofexternal rays, the proposition below must be true.28

1.2 Proposition. Let (F, J ) be an admissible critical portrait and Γ a finite invariantset of special arguments.

With the notation above, J ∗is weakly unlinked to F∗in the right.The reader can skip the rest of this section without any loss of continuity. The proofof the proposition follows immediately from Lemmas 1.3-1.9.1.3 Lemma.

Suppose θ1 ≈θ2, ψ1 ≈ψ2 but θ1 ̸∼l ψ1. Then {θ1, θ2} and {ψ1, ψ2} areunlinked.Proof.

Suppose this is not the case. We assume then that {θ1, θ2} and {ψ1, ψ2} arelinked because θ2 = ψ2 implies θ1 ∼l ψ1.As a preliminary remark suppose A−(θ1) =A−(θ2) = A−(ψ1) = A−(ψ2); then as the cyclic order of these elements is preserved by md(compare Lemma II.1.1), {md(θ1), md(θ2)} and {md(ψ1), md(ψ2)} are still linked.

For theproof we distinguish several cases.Case 1: S−(θ1) = S−(θ2) and S−(ψ1) = S−(ψ2). This possibility is easily ruled outusing Lemma II.1.4.

We can say even more. If A−(θ1) = A−(θ2) and A−(ψ1) = A−(ψ2) thenby that same lemma we have also A−(θ1) = A−(ψ1).

Thus, according to our preliminaryremark, it is enough to consider the case when A−(θ1) ̸= A−(θ2).Case 2: θ1, θ2 ∈Jk.As ψ1 and ψ2 belong to different components of R/Z −Jk,by definition A−(ψ1) ̸= A−(ψ2). Thus, also by definition S−(ψ1) ≡i S−(ψ2) for some i.But then, again by definition, there are ψ′j ∈Ji (j = 1, 2), each in the same connectedcomponent of R/Z −{θ1, θ2} as ψj, with S−(ψ′j) = S−(ψj).

But this is a contradictionwith the fact that Jk, Ji are unlinked.Case 3: S−(θ1) ≡k S−(θ2) and S−(ψ1) = S−(ψ2). By definition, there is θ′1 ∈Jk suchthat S−(θ′1) = S−(θ1).

Now, if θ1 and θ′1 belong to different components of R/Z −{ψ1, ψ2}then {θ1, θ′1}, and {ψ1, ψ2} are linked and we are in case 1. Otherwise, we repeat the samereasoning using now θ2 and we reach either case 1 or case 2.Case 4: S−(θ1) ≡k S−(θ2) and S−(ψ1) ≡j S−(ψ2).

We proceed as in case 3 and thisis reduced to either case 2 or case 3.#1.4 Corollary. The ∼l equivalence classes are unlinked.#1.5 Lemma.

For any Fk ∈F and any ∼l equivalence class Λ, {Λ} is weakly unlinkedto {Fk} in the right.Proof.Let θ0 ∈Λ and take γ1, γ2 consecutive in Fk so that θ0 ∈(γ1, γ2].It isenough to prove that if θ0 ≈θ1 then also θ1 ∈(γ1, γ2]. If A−(θ0) = A−(θ1), this follows bydefinition (θ0 and θ1 by definition belong to the same connected component of R/Z −Fk).So suppose that S−(θ0) ≡i S−(θ1) with θ1 ̸∈(γ1, γ2].

In this case there exist Ji ∈J so that29

θ′0 ∈Ji ∩(γ1, γ2] and θ′1 ∈Ji ∩(γ2, γ1] with S−(θj) = S−(θ′j). But this is a contradictionwith condition (c.2) in the definition of critical portraits (Ji will not be weakly unlinked toFk in the right).#1.6 Lemma.

Let ψ1 ∼γ ψ2 and γ ̸∈Fk, then {ψ1, ψ2} and Fk are unlinked.Proof. If A+(ψ1) = A+(ψ2) this follows by definition and Remark II.1.7.

Otherwisewe must have that γ ∈Fi for some i ̸= k. But then a similar argument as that used inLemma 1.5 shows that Fi and Fk are not unlinked.#1.7 Lemma. Let θi ∼γi ψi, i = 1, 2 with γ1 ̸= γ2.

Then {θ1, ψ1} and {θ2, ψ2} areunlinked.Proof. We will consider right symbol sequences S+(θj) and S+(ψj).

Suppose is notthe case that they are unlinked. Then {θ1, ψ1} and {θ2, ψ2} are linked because θ2 = ψ2will imply γ1 = γ2 by Lemma II.4.3.

As preliminary remarks, suppose A+(θ1) = A+(ψ1) =A+(θ2) = A+(ψ2). Then as the cyclic order of these elements is preserved by md (com-pare Lemma II.1.1), {md(θ1), md(θ2)} and {md(ψ1), md(ψ2)} are linked.

Furthermore, ifA+(θ1) = A+(θ2) and A+(ψ1) = A+(ψ2), by Lemma II.1.4 we must have A+(θ1) = A+(ψ1).Now, suppose θ1 is in the same connected component of R/Z −{θ2, ψ2} as γ1 (if notψ1 will be). In this case {θ′1 = γ1, ψ1}, and {θ2, ψ2} are linked, so we assume θ1 = γ1.

Inan analogous way we may suppose that θ2 = γ2. Under this assumption we will prove thatfor all j ≥0, {m◦jd (θ1), m◦jd (ψ1)} and {m◦jd (θ2), m◦jd (ψ2)} should be linked.

Of course thisis absurd because by definition, for j big enough we have m◦jd (θ1) = m◦jd (ψ1) = m◦jd (γ1).Suppose that A+(θ1) ̸= A+(ψ1). Then by definition θ1 ∈Fk for some k. Furthermore,there is ψ′1 ∈Fk with A+(ψ′1) = A+(ψ1).

It follows from Lemma 1.6 that θ1, ψ′1 ∈Fk are inthe same component of R/Z −{θ2, ψ2}. Thus, {ψ′1, ψ1} and {θ2, ψ2} are still linked.

Notethat md(ψ′) = md(θ1). Also by symmetry we may take A+(θ2) = A+(ψ2) (note that theproperty md(θ2) = md(γ2) will not be lost).

But then by the second preliminary remarkA+(θ1) = A+(ψ1) = A+(θ2) = A+(ψ2), and so, by the first {md(θ1) = md(γ1), md(ψ1)}and {md(θ2) = md(γ2), md(ψ2)} are linked. This is the desired contradiction.#1.8Corollary.Thefamily{{θ:θ∼γγ}:γ∈O(F∪per)}isunlinked.#1.9 Lemma.Let γ ∈O(F∪per) and Λ an ∼l equivalence class.Then Λ is weaklyunlinked in the right to any finite subset of {θ : θ ∼γ γ}.Proof.

Take γ ∈Fγ ∈F. We will prove by induction that any ∼l equivalence classΛ, is weakly unlinked to Ψn(γ′) = {θ ∼γ′ γ′ : m◦nd (θ) ∈Fγ} (here γ′ belongs to the samecycle as γ, and m◦nd (γ′) = γ).

The result follows easily. For n = 0, this is Lemma 1.5.30

In general take θ1 ≈θ2 and assume that {θ1, θ2} is not weakly unlinked in the right to{ψ1, ψ2} ⊂Ψn(γ′).Case 1: A+(ψ1) ̸= A+(ψ2). Then by definition γ′ ∈Fk ∈F for some k. Thus, thereare ψ′i ∈Fk such that A+(ψ′i) = A+(ψi), and because of Lemma 1.5, it is easy to see that{θ1, θ2} is not weakly unlinked in the right to either {ψ1, ψ′1} or to {ψ2, ψ′2} (both beingsubsets of Ψn(γ′)).

Thus it is enough to consider case 2.Case 2: A+(ψ1) = A+(ψ2).In this case we can not have simultaneously θ1 = ψ1and θ2 = ψ2.In fact, in this case Lemma II.1.1 would imply that {md(θ1), md(θ2)} isnot weakly unlinked in the right to {md(ψ1), md(ψ2)} in contradiction with the inductivehypothesis. Thus we may suppose that θ1 ∈(ψ1, ψ2) (and θ2 ∈(ψ2, ψ1]).

If A−(θ1) =A−(θ2) it follows from Lemma II.1.4 that for ǫ > 0 small enough A+(θ1 −ǫ/d) = A+(θ2 −ǫ/d) = A+(ψ1) = A+(ψ2). By Lemma II.1.1 we have then that {md(θ1) −ǫ, md(θ2) −ǫ}and {md(ψ1), md(ψ2)} are not unlinked, in contradiction with the inductive hypothesis.Therefore A−(θ1) ̸= A−(θ2), and then by definition we must have S−(θ1) ≡i S−(θ2).

Butthen, using the same reasoning as in the previous lemmas, we can assume that θ1, θ2 ∈Ji.But if this is the case, we get a contradiction because it follows by definition and RemarkII.1.7 that A+(ψ1) ̸= A+(ψ2).#Proposition 1.2 follows now easily from the above lemmas.#2. Abstract and embedded webs.In this section we construct from the combinatorial data a topological polynomial ofdegree d. We also study some of its basic properties.

None of the material presented hereis essentially new, and can be found in a slightly different formulation in [BFH].2.1 Let (F, J ) be an admissible critical portrait. For any finite invariant set of specialarguments Γ, we consider the pair (F∗, J ∗) as in Section 1.

With these families, we constructfirst an abstract topological graph W(F∗, J ∗) as follows. We pick a vertex v = ∞, and takeas many edges Rθ incident at ∞as elements θ ∈J ∗∪.

Let vθ be the other adjacent vertexto Rθ. We identify the vertices vθ, vθ′ if and only if θ, θ′ ∈J ∗k for some k; that is, if andonly if θ ∼l θ′.

(This because we are expecting the rays with arguments ∼l related to landat the same point.) We write this vertex as v(J ∗k ).

As each Rθ is labeled by an argumentθ, we call it the web ray of argument θ. By abuse of language we will say that vθ (= v(J ∗k )whenever θ ∈J ∗k ) is the landing point of the web ray Rθ.Next, for each subset F∗k ∈F∗we consider a new vertex ω(F∗k).

We join this vertexto the landing points of Rγ for all γ ∈F∗k. (This because, all those rays are supposed tosupport the same Fatou component; compare Proposition II.4.7).

In this case the extendedweb ray Eγ is the set formed by the web ray of argument γ, its landing point, and theedge joining this landing point with the vertex ω(F∗k). In each set F∗k ∈F∗∪there is a31

preferred argument γk. We call the edge ℓF ∗k joining ω(F∗k) with vγ, the preferred internalray associated with the “Fatou type” point ω(F∗k).Note that by construction (compare §1.1), the argument 0 is always present in ourconstruction.We say that the web ray R0 is the preferred internal ray associated withv = ∞.

The graph W(F∗, J ∗) constructed in this way, is the abstract web associated with(F, J , Γ). We will denote by V the set of vertices of this graph.2.2 Embedded webs.

We consider embeddings in the Riemann Sphere ˆC of thisabstract web W = W(F∗, J ∗). An embedding such that the cyclic order of the web rayscorresponds to the cyclic order of the labeling by arguments can always be constructed be-cause of Proposition 1.2.

We can always assume that the respective points at ∞correspond.Any such embedding is an embedded web. We still call the image of edges incident at “∞”web rays.

Unless strictly necessary we will not distinguish between an embedding and itsimage.32

2.3 Web maps. The following two properties follow immediately from the constructionof (F∗, J ∗) and Lemmas II.3.2 and II.4.5.If θ, θ′ ∈J ∗k , there is a unique J ∗f(k), such that md(θ), md(θ′) ∈J ∗f(k).If γ, γ′ ∈F∗k, there is a unique F∗f(k), such that md(γ), md(γ′) ∈F∗f(k).These two conditions allow us to define a map f between the set vertices of the webW(F∗, J ∗) (also define f(∞) = ∞).We can extend this map to a map of the wholegraph W(F∗, J ∗) as follows.For any edge which is a web ray Rθ, define f|Rθ as anhomeomorphism between this edge and the web ray Rmd(θ).

Otherwise, if ℓwith adjacentvertices v1, v2 is not a web ray, define f|ℓas an homeomorphism between this edge and theunique edge with adjacent vertices f(v1), f(v2).Note that the above construction determines intrinsically the concept of periodic andpreperiodic edges in the web. Also note that preferred internal rays map to preferred internalrays.Next, we consider an embedding φ : W = W(F∗, J ∗) →ˆC.

Any web map f induces amap ˆf of W = φ(W) to itself by the formulaˆf(z) = φ(f(φ−1(z))).By a regular extension of ˆf will be meant any extension of ˆf which is a degree d orientationpreserving branch map of the extended complex plane. Keeping track of the embeddedvertices this extension is essentially unique.2.4 Theorem.

Let φ1, φ2 be two embeddings of the abstract web W = W(F∗, J ∗).Let ˆfi : ˆC →ˆC (i = 1, 2) be regular extensions of the web maps. Then ( ˆf1, φ1(V)) and( ˆf2, φ2(V)) are Thurston equivalent as topological maps (compare Appendix A).In fact, there are homeomorphisms ψα, ψβ : ˆC →ˆC isotopic relative to φ1(V) so thati) For every vertex v ∈V, ψα(φ1(v)) = ψβ(φ1(v)) = φ2(v).ii) The diagramˆCψβ−→ˆCˆf1yyˆf2ˆCψα−→ˆCis commutative.Proof.

It is not difficult and can be found in [BFH, Theorem 6.8].#2.5 Lifting Webs. Suppose W = φ(W(J ∗, F∗)) is an embedded web.

Given thisembedding, we fix a regular extension ˆf : ˆC →ˆC of the web map. If W′ is another embedded33

web isotopic to W relative to the set φ(V), then ˆf uniquely determines an embedded webW′′ ⊂ˆf −1(W′) which is also isotopic to W relative to φ(V), as the following constructionshows.It is convenient first to define “the web ray of argument 0” in W′′. For this we needthe following remark.Let θ ̸= 0 belong to J ∗∪.

If 0 ∼l θ, then the web rays Rθ and R0 in W can not beisotopic relative to the set φ(V).To see this we note that these web rays determine two sectors. By construction each ofthese two sectors contains all web rays with arguments in (0, θ) and (θ, 1) respectively.

Now,by Lemma II.1.6, θ is of the form k/(d −1), so each of the sets J ∗∪∩(0, θ) and J ∗∪∩(θ, 1)is non empty. The result follows easily.As a consequence we have that there is a unique edge R′0 in W′ which can correspondto R0.

Thus there is a unique ‘edge’ R′′0 ⊂ˆf −1(R′0) joining φ(v0) and ∞, which is isotopicto R0 relative φ(V). This is to be defined as the zero web ray in W′′.To construct the web W′′ we consider first all edges ℓ⊂W incident at vertices v =φ(v(J ∗k )) which are not critical.

By definition, ˆf(ℓ) is also an edge in W; now, there is aunique edge ℓ′ ∈W′ which is isotopic to ˆf(ℓ) relative to φ(V). As ˆf is locally one to onenear v, starting at ˆf(v), ℓ′ can be lifted back in a unique way by ˆf to an arc ℓ′′.

As ˆf(ℓ)and ℓ′ are in particular isotopic relative to the critical values of ˆf, it follows that ℓand ℓ′′are isotopic relative to φ(V).Finally, we consider all edges ℓincident at critical vertices v = φ(v(J ∗k )). Again werepeat the same procedure but keeping in mind that the correct indexing for web rays canbe found by its relative position respect to the web ray R0.

The adequate choice of inversescan now be easily determined. This finishes the construction of W′′.

By abuse of notation,we denote this embedded web W′′ by ˆf −1(W′).Note that we can apply the same construction to the web W′′ = ˆf −1(W′) and so on;in this way we can form a sequence of websW′, ˆf −1(W′), . .

. , ˆf −n(W′), .

. .all isotopic relative to φ(V).3.

There are no Levy cycles.In this Section we will prove that any admissible critical portrait is ‘naturally’ associatedto a unique polynomial P (see Corollary 3.6). The natural way to proceed is to construct34

from the family (F∗, J ∗) with Γ = ∅a web map ˆf. The next step can be (as in [BFH])to prove that any regular extension has no Thurston’s obstruction by proving there are noLevy cycles.

This fact is by no means obvious. In fact, it is easier to prove this fact formaps ˆf ′ associated to a bigger family (F′∗, J ′∗) with Γ suitably chosen.

Now, as a Levycycle for the map ˆf will determine a Levy cycle for the map ˆf ′ we can conclude that theformer map has no Levy cycles.We start with some notation and another result borrowed from [BFH] Section 7.3.1 Definition. Let W be an embedded web and ℓ⊂W an edge.

A Jordan curveC disjoint from φ(V) is said to intersect ℓessentially, if for every C′ homotopic to C inˆC −φ(V), we have that ℓ∩C′ is non empty.The following is together with Theorem A.5 a technical result needed for the proof ofthe main theorem.3.2 Lemma. Suppose ˆf admits a Levy cycle Λ = {C1, .

. .

, Ck} (see appendix A). Thenany Ci does not intersect a preperiodic edge ℓof the web in an essential way.Proof.

See [BFH] Lemma 7.7.#3.3 Remark. Using Proposition II.4.6 it is easy to construct a finite set of specialarguments Γ with the following properties.i) md(Γ) ⊂Γ ∪O(F∪).ii) If λ ∈F∪per, and λ′ is the successor (counterclockwise) of λ in J ∗∪then λ ∼λ λ′.In the following lemma we assume that the web and a regular extension where con-structed with this set of special arguments.

Here if C is a Jordan curve, the interior of C isdefined as the bounded component of ˆC −C.3.4 Lemma. Let C be a Jordan curve disjoint from φ(V).

Suppose further that C hasthe following properties,a) All vertices in φ(V) which belong to the interior of C are periodic and do not belongto a critical cycle.b) C does not intersect essentially any preperiodic edge ℓ.Under theses hypothesis, if vθ, vθ′ ∈φ(V) (corresponding to the landing point of theweb rays Rθ, Rθ′ respectively) belong to the interior of C, then A−(θ) = A−(θ′).35

Proof. Suppose vθ, vθ′ are in the interior of C (and therefore θ, θ′ are periodic).

Letγ, γ′ ∈Jk for some k. The rays Rγ and Rγ′ divide the plane in two regions. If vθ, vθ′ do notbelong to the same region, then C will cut either Rγ or Rγ′ in an essential way.

Thus, θ, θ′belong to the same connected component of R/Z−Jk. Now, let γ, γ′ ∈Fk for some k. Theextended rays Eγ and Eγ′ divide the plane in two regions.

If both γ, γ′ are preperiodic thesame argument as above applies, and again θ, θ′ belong to the same connected component ofR/Z−{γ, γ′}. Otherwise, suppose that γ is periodic (and thus, γ′ must be preperiodic).

Byhypothesis there is ǫ > 0 such that γ +ǫ is a special argument for γ, and (γ, γ +ǫ)∩J ∗∪= ∅.Now we apply the same reasoning with the extended rays Eγ+ǫ and Eγ′ and thus θ, θ′ belongto the same connected component of R/Z −{γ + ǫ, γ′} (compare Figure 3.1). As ǫ can bechosen arbitrarily small, it follows that for ǫ > 0 small enough, θ −ǫ and θ′ −ǫ belong tothe same connected component of R/Z −Fk.

It follows by definition that A−(θ) = A−(θ′).#3.5 Proposition. Let ˆf : ˆC →ˆC be a regular extension of the web map over (F∗, J ∗)for some Γ.

Then ˆf admits no Levy cycles.Proof. We are going to add points to Γ as needed (see the introduction to this section).Suppose by contradiction that ˆf has a Levy cycle {C1, .

. .

, Ck}.Step 1. As all “Fatou points” (i.e, vertices of the form φ(ω(F∗j ))) are preperiodic orbelong to a critical cycle, no such points are in the interior of an element of a Levy cycle(compare Theorem A.5).Step 2.

If θ ∼l θ′ but S−(θ) ̸= S−(θ′) then θ is preperiodic, and so is vθ. Thus, vθ isnot in the interior of a curve in a Levy cycle.Step 3.

If vθ, vθ′ are in the interior of an element of a Levy cycle, then by Lemma 3.4A−(θ) = A−(θ′).Step 4. There are no Levy cycles:If vθ, vθ′ belong to the interior of an element C1 of a Levy cycle, then there is anotherelement C in this Levy cycle such that vmd(θ) and vmd(θ′) belong to the interior of C. Thisimmediately implies S−(θ) = S−(θ′) by step 3 and the definition of Levy cycles.

In this wayvθ = vθ′ by construction of the Web. But this implies there is a unique point in the interiorof an element of a Levy cycle, and this is a contradiction with the definition of Levy cycles.#3.6 Corollary.

Let (F, J ) be an admissible critical portrait. There is a unique (up toconjugation) polynomial P(F, J ) which is Thurston equivalent to ˆf.

Here ˆf is any regularextension of the web map.#3.7 Example. We are left with the awkward situation of illustrating a result aboutthe impossibility of Levy cycles.

In order to do this, some hypothesis must be violated. We36

have chosen to violate the condition which avoids the existence of Levy cycles, namely that∼l equivalence classes determine only one point in the Julia set.We consider the admissible critical portrait F = {{ 14, 712}, { 34, 112}} and J = ∅(compareexample I.4.4). It is easy to check that S−( 14) = S−( 34) (thus expecting the rays R 14 and R 34to land at the same point in the Julia set).

We consider also the set of special argumentsΓ = { 1336, 3136} which satisfies the hypothesis stated in 3.3 (here 1336 ∼1414 and 3136 ∼3434). Thuswe have formedF∗= {{14, 1336, 712}, {34, 3136, 112}}J ∗= {{0}, { 112}, {14, 34}, {1336}, { 712}, {3136}}(recall the meaning of the elements in each family).To illustrate Lemma 3.4 (and Proposition 3.5), we construct a web W(F∗, J ∗) withoutidentifying v 14 and v 34 .

We will show how this leads to a Levy cycle (compare Figure 3.1).Lemma 3.4 claims that if there is a Levy cycle, then arguments of any two vθ, vθ′ inthe interior of a constituent element C of this cycle should have the same left address. Inour case this means that any such C can not cross any solid segment in Figure 3.1 becauseof Lemma 3.2.

Thus, the only possibility of a cycle is as shown in Figure 3.1. Of course,with the appropriate identification of v 14 and v 34 , this is impossible.Figure 3.14.

Untwisting the conjugacy.Up to this point Corollary 3.6 tells us there is a polynomial (unique up to conjugation)associated with the admissible critical portrait (F, J ). We must still prove that externaland internal rays land at the expected places.

In other words, we have to prove that suchpost-critically finite polynomial admits the required marking. The proof of this fact is notas obvious as it will seem.

We will consider first a particular example in order to show whichdifficulties we can still find and describe a way to handle them.37

4.1 Example. Consider the admissible critical portrait formed with F = {{0, 13, 23}},J = ∅.

We first look at the map f(z) = z3 as a ‘topological polynomial’ in the web W(F, J )with vertices V = {0, 1, e2πi3 , e4πi3 } and extended web rays Ek/3 = {re2kπi3: r ∈[0, ∞)}for k = 0, 1, 2.By Corollary 3.6 this topological polynomial is equivalent to a uniquepolynomial, which will surely be P(z) = z3.Consider the homeomorphismsψ0(r3e2πiθ) =r3e2θπiif r ≤3;r3e2πi[θ+ 32 ( lnr−ln3ln4−ln3 )]if 3 ≤r ≤4;r3e2πi[θ+ 32 ]if 4 ≤r.ψ1(re2πiθ) =re2θπiif r ≤3;re2πi[θ+ 12 ( lnr−ln3ln4−ln3 )]if 3 ≤r ≤4;re2πi[θ+ 12 ]if 4 ≤r.Then clearly the following diagram is commutativeˆCψ1−→ˆCfyyP.ˆCψ0−→ˆC(1)We describe what is happening in the following terms. The map ψ0 makes a ‘Dehntwist’ of 3/2 turns far from ∞.

Thus the ‘Web’ ψ0(W(F, J )) itself is twisted 3/2 turns.By this we mean that when keeping track of the image ψ0(R0) of the web ray R0, we startas the actual ray R0 for a while, then twist in counterclockwise direction until we havecompleted 3/2 turns, and finally continue our way to ∞following the ray R1/2! Similarlywith all other web rays.Now, when lifting back the web ψ0(W(F, J )) by P −1 (compare §2.6), we see that theresulting embedded web ψ1(W(F, J )) has a completely different behavior (but they areisotopic).

The image web ray ψ1(R0) in this case goes for a while in the direction of theactual ray R0, then twists 1/2 turns, and finally continues in the direction of the actual rayR1/2 to ∞.The situation is even worse if we consider successive liftings of the web ray ψ0(R0). Inthese cases, near ∞they will be successively identified with the rays R 12 , R 16 , R 118 , .

. .. Ofcourse, we will prefer to have always near ∞the correct identification.

In order to describea possible solution to this dilemma, we note that ψ0(z) = ψ1(z) for |z| big enough. If weremove the set {z : |z| > α} for α big enough, ψ0 and ψ1 would not be isotopic in thisnew Riemann surface relative to the boundary (they will differ by exactly ‘one turn’ around{z : |z| = α}.This is hardly a surprise because the difference in 1 turn can be easilymeasured by comparing the embedded web to its lift.

Now, it is clear that we have notstarted with the best possible choice of a web. Our original web was ‘twisted’ by a givennumber of turns (3/2 in this case); when we ‘lift back’ the web, this twist will be divided38

by the degree of the polynomial (3 in this case). Thus, the ‘difference in twist’ (which canalways be measured) allows us to state the relationtwist −twistd= difference in twist.

(2)Where d is the degree of the polynomial (here d = 3) and difference in twist is the relativetwist of the ray ψ1(R0) in the lifted web respect to the original ψ0(R0). In this way, equation(2) suggests that any possible odd behavior when lifting webs is because of a ‘Dehn twist’in a neighborhood of Fatou points.

This is going to be in general the case as we will showbelow.4.2. In the general case, we have that starting from the admissible critical portrait(F, J ) we can construct a unique up to conjugation polynomial P of degree d (which wetake here to be monic and centered).

Also diagram (1) holds. Furthermore, by replacing fby ψ0 ◦f ◦ψ−10and ψ1 by ψ1 ◦ψ−10 , we may assume without loss of generality that ψ0 = id.For notational convenience we include ∞in the critical set Ω(P) of the polynomial P.For each periodic Fatou point ω ∈Ω(P), let φω denote a fixed B¨ottcher coordinate associatedwith ω (∞included).For r < 1 define Nr(w) = {z ∈U(ω) : |φω(z)| < r}.

For eachstrictly preperiodic Fatou point c ∈O(Ω(P)), we inductively define Nr(c) as the connectedcomponent of P −1(Nr(P(c))) containing c. For X ⊂O(Ω(P)) set Nr(X) = ∪c∈XNr(c).Now, as there is no topological way to distinguish between the sets ˆC −O(Ω(P)) andˆC −Nr(O(Ω(P))), we can construct an embedded web in ˆC and a regular extension f suchthat the following conditions are satisfied,i) f = P in N1/2(O(Ω(P))),ii) preferred internal web rays are equal to internal preferred rays in N1/2(ω) if ω is ina critical cycle, andiii) Web edges correspond to internal rays in N1/2(O(Ω(P))).Denote by W the so constructed web, and by V be the respective set of vertices (thereis no further need to write this set as φ(V)). Recall we are assuming that ψ0 is the identityin diagram (1).

Note also that the construction implies that near periodic critical points,ψ1 is a rotation in the B¨ottcher coordinate.4.3 Untwisting external rays. We consider first what happens near ∞(for example,in the set N1/2(∞)).

As diagram (1) is commutative, we have that for any positive r ≤1/2,ψ0(W) ∩Nr(∞) is by construction ∞and some segments of actual external rays.Theportion of the web ray ψ1(R0) ∩Nr(∞) must then be a segment of a ray of the form Rj/d.Furthermore, we can measure the relative twist of ψ1(R0) respect to ψ0(R0) in ∂Nr(∞)(which by construction is a rational number of the form k/d). Stating this as an equationpossible twist −possible twistd= difference in twist39

we have necessarily a rational solution of the form k/(d −1) (same k as above).To prove that this ‘possible twist’ is in fact a twist we proceed as follows.Take apositive s < r and consider the annulus Nrd(∞) −Nsd(∞). We modify ψ0 in Nrd(∞) bymaking a twist of −kd−1 turns inside this annulus.

This forces us to modify ψ1 in Nr(∞) bya twist of −kd(d−1) turns inside the annulus Nr(∞) −Ns(∞) in order to make diagram (1)commutative. Clearly there is no problem in doing so because ψ0 is the identity in Nrd(∞),and ψ1 is a rotation in the set Nr(∞) respect to the B¨ottcher coordinate.Formally, we have that in the set ˆC −V −Nr(∞), ψ0 and ψ1 are not isotopic respectto the boundary because they differ by k/d turns.

In the annulus Nr(∞) −Nsd(∞), themodified ψ0, ψ1 differ by −k/d turns. In this way, the modified ψ0, ψ1 are isotopic relative tothe boundary in ˆC−V−Nsd(∞).

Thus, the ‘difference in twist’ between the ‘new’ web raysψi(R0) is 0 when measured in ∂Nsd(∞). In particular, if we consider the successive liftingof webs P −n(ψ0(W(F∗, J ∗))) (compare §2.5), all these webs (by construction) will have nodifference in twist between the respective lifts of web rays of argument 0.

We remark thatnear ∞those web rays are now identified with the ray R−k/d−1. Also the respective liftingof web rays correspond to bigger and bigger portions of actual rays.

Of course these raysdo not necessarily correspond to the expected ones, but they will after conjugation of thepolynomial P with A(z) = e−2kπid−1 z.4.4 Untwisting periodic preferred internal rays. Our next step is to make theanalogue construction in the basin of attraction of finite periodic critical cycles.

Supposeω0 7→ω1 7→. .

. 7→ωn = ω0 is a critical cycle, and let di be the local degree at ωi.

Thecritical cycle has total degree D = d0 × . .

.× dn−1. Under the same philosophy as in §4.3 wewill like to prove that each coordinate in this cycle was ‘twisted’ by say xi turns.

We willdenote by ℓi the preferred internal web edge adjacent to ωi.What we can surely do, is to measure the difference in twist when we lift back webs.In other words the relative twist of ψ1(ℓi) ⊂P −1(ℓi+1) respect to ψ0(ℓi). Let this valuebe yi (which by construction is a rational number with denominator di).

If it is true thatthe coordinates are ‘twisted’, then the ‘possible twist’ of ψ0(ℓi) is by construction xi; whilewhen ‘lifting back’ ℓi+1 to get ψ1(ℓi), its ‘possible twist’ xi+1 is divided by di. Thus, if wewant to proceed as in §4.3 we must be able to solve the system of equationsxi = xi+1di+ yii = 0, .

. .

, n −1for xi rational with denominator D −1. But it is clear that this can be done if we rewritethe system asd0d1 .

. .

dn−1x0=d1 . .

. dn−1x1+d0d1 .

. .

dn−1y0d1 . .

. dn−1x1=d2 .

. .

dn−1x2+d1 . .

. dn−1y1.........dn−2dn−1xn−2=dn−1xn−1+dn−2dn−1yn−2dn−1xn−1=x0+dn−1yn−140

With the given solutions x0, . .

. , xn−1 we proceed to untwist the conjugacy in all neigh-borhoods of the cycle simultaneously as in §4.3.4.5 Untwisting non periodic Fatou critical components.

The last basins thatneed to be ‘untwisted’ are the ones that correspond to strictly preperiodic Fatou criticalpoints. Let ω be such critical point, and ω′ = f ◦n(ω) the first critical point in its forwardorbit.

We assume that near ω′ the conjugacy has been already ‘untwisted’. In this case theresulting equation is simply xω = yω so we proceed again as in §4.3.5.

Proof of Theorem I.3.9.5.1 Now we apply successively the construction in §2.5. The webs Wn = P −n(ψ0(W))have edges which coincide with the actual internal and external rays in a bigger set aftereach lifting.

Given n, for the web Wn we consider for each ¯v landing point of “web rays”and for each edge ℓincident at it, the orbifold length of ℓn = ˆC −Nr−dn(O(Ω(P))) ∩ℓ. Forfixed n denote by δn the supremum of such numbers over all possible vertices and edges.Note that, as the orbifold metric is strictly expanding for P in ˆC −Nr(O(Ω(P))), and eachℓn is the inverse image of some ℓ′n−1 we have that δn ↓0.

In this way we have that therespective rays (internal and external) of P can be found arbitrarily close to the expectedlanding points. As J(P) is locally connected they actually land there.5.2 To finish the proof of the theorem, we only have to prove that the rays Rγ associatedwith a Fatou periodic critical point actually support the respective component.

But this istrivial if we consider Proposition II.4.6. In this case Rγ, Rγ+ǫ land in the boundary of thesame critical component (compare Proposition II.4.7).

Thus, in the region determined bythe extended rays ˆRγ, ˆRγ+ǫ there is no place for a periodic ray Rλ of the same period asRγ, if ǫ > 0 was chosen small enough. This completes the proof of Theorem I.3.9.#41

Appendix AThurston’s Topological Characterization of Rational Maps.Let f : S2 7→S2 be an orientation preserving branched covering map of the topologicalsphere. The set Ω(f) of all critical points of f is called the critical set of f. The postcriticalset of f is the set P(Ω(f)) = S∞n=1 f ◦nΩ(f).

Whenever the set P(Ω(f)) is finite, we say thatf is postcritically finite.In what follows, we assume always that f is postcritically finite. A finite invariant setM; i.e, f(M) ⊂M, containing all critical points of f is called a marked set.

In analogywith the previous notation, we set P(M) = S∞n=1 f ◦nM, and call it a postmarked set.The elements of M (respectively P(M)) are called marked points (respectively postmarkedpoints). We say that (f, M) is a marked branched map.Twomarkedbranchedmaps(f, M(f))and(g, M(g))areThurstonequivalent if there are homeomorphisms φ1, φ2 : S2 →S2, isotopic relative to the setP(M(f)) such that g ◦φ1 = φ2 ◦f, and φ1(P(M(f))) = φ2(P(M(f))) = P(M(g)).We say that a simple closed curve γ ⊂S2 −P(M) is non-peripheral (for the markedbranched map (f, M)), if each component of S2 −γ contains at least two points of P(M).

Amulticurve Γ = {γ1, ..., γn} is a set of simple, closed, disjoint, non-homotopic, non-peripheralcurves in S2 −P(M). A multicurve Γ is stable, if for every γ ∈Γ, every non-peripheralcomponent of f −1(γ) is homotopic (relative to P(M)) to a curve in Γ.Let γi,j,α be the components of f −1(γj) homotopic to γi relative to P(M), and di,j,αbe the degree of the map f|γi,j,α : γi,j,α 7→γj.

We define the (i, j) entry of the ThurstonMatrix fΓ as(fΓ)i,j =X1/di,j,α.Note that by the Perron-Frobenious theorem there is a largest positive eigenvalue λ(fΓ).There is a smaller function ν : PM 7→{1, 2, ..., ∞}, such that for all x ∈f −1(y), ν(y)is a multiple of ν(x)degxf. We have that the orbifold (S2, PM, νf) is hyperbolic if its “Eulercharacteristic” satisfies2 −Xx∈P (M)(1 −1/νf(x)) < 0.Note that we are allowing extensions of the critical and postcritical sets.

This is because wewhat to use Thurston’s theorem in more generality than presented in [DH2] and used in [F]42

or [BFH]. Our marked set is the usual one and maybe a finite number of additional periodicor preperiodic orbits.

Note that at these additional points, the orbifold function has value1, so that the orbifold structure is only determined by the original postcritical set P(Ω(f)).A.1 Theorem (Thurston’s Characterization of Rational Maps).A markedbranched map, with hyperbolic orbifold is equivalent to a rational function if and only if forany stable multicurve Γ, we have λ(fΓ) < 1. In this case the rational function is unique upto conjugation by a Mobius transformation.Proof.

The proof in [DH2] applies without modification.#A.2 Topological Polynomials. A branched map f : S2 7→S2 is said to be a topo-logical polynomial if f −1(∞) = ∞.If we are interested only in topological polynomials Thurston’s theorem is equivalentto the following (see [BFH Theorem 3.2]).A.3 Theorem.

A marked topological polynomial (f,M) is equivalent to a polynomial ifand only if for any stable multicurve Γ we have λ(fΓ) < 1. In this case, the polynomial isunique up to conjugation by an affine transformation.Definition.A stable multicurve Γ, with λ(fΓ) ≥1 is called a Thurston Obstruction(for (f,M)).Levi CyclesEverything here is taken from [BFH] section 4.Let (f, M) be a marked topological polynomial.

Let Γ be a stable multicurve. Supposethere exists {γ0, ..., γk = γ0} = Λ ⊂Γ such that for each i = 0, ..., k −1, γi is homotopicrelative to P(M) to exactly one component γ′ of f −1(γi+1).

Suppose also that f : γ′ 7→γi+1has degree 1. Then Λ is called a Levy cycle.A.4 Theorem.

If a marked topological polynomial (f,M) has a Thurston obstructionΓ, then (f,M) has a Levy cycle.A.5 Theorem. The disks of the elements of Λ = {γ0, ..., γk = γ0} (i.e, the boundedcomponents of S2 −γi), contain only cycles of periodic non-critical points of P(M).43

The last two Theorems together have an interesting interpretation.For Post-critically finite topological Polynomials, only misidentification of periodic pointscan lead to an obstruction.44

References. [BFH] B. Bielefeld, Y.Fisher, J. Hubbard, The Classification of Critically PreperiodicPolynomials as Dynamical Systems; Journal AMS 5(1992)pp.

721-762. [DH1] A.Douady and J.Hubbard, ´Etude dynamique des polynˆomes complexes, part I;Publ Math.

Orsay 1984-1985. [DH2] A.Douady and J.Hubbard, A proof of Thurston’s Topological Characterizationof Rational Maps; Preprint, Institute Mittag-Leffler 1984.

[F] Y.Fisher, Thesis; Cornell University, 1989. [GM] L.Goldberg and J.Milnor, Fixed Point Portraits; Ann.

scient. ´Ec.

Norm. Sup.,4e s´erie, t. 26, 1993, pp 51-98[M] J.Milnor, Dynamics in one complex variable:Introductory Lectures; Preprint#1990/5 IMS SUNY@StonyBrook.

[P]A.Poirier,OnPostcriticallyFinitePolynomials;Thesis,SUNY@StonyBrook 1993.45

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