Infinite-dimensional Thurston theory and transcendental dynamics II: classification of entire functions with escaping singular orbits
aa r X i v : . [ m a t h . D S ] F e b INFINITE-DIMENSIONAL THURSTON THEORYAND TRANSCENDENTAL DYNAMICS II:CLASSIFICATION OF ENTIRE FUNCTIONSWITH ESCAPING SINGULAR ORBITS
KONSTANTIN BOGDANOV
Abstract.
We classify transcendental entire functions that are compositionsof a polynomial and the exponential for which all singular values escape ondisjoint rays. The construction involves an iteration procedure on an infinite-dimensional
Teichm¨uller space, analogously to the Thurston’s classical Topo-logical Characterization of Rational Functions, but for an infinite set of markedpoints.
Contents
1. Introduction 12. Escaping set of functions in N d
43. Setup of Thurston iteration and contraction 104. Id-type maps and spiders 115. Invariant compact subset 146. Appendix A 237. Acknowledgements 25References 25This is the second out of four articles publishing the results of the author’sdoctoral thesis [Bthe]. The machinery needed for the Thurston’s iteration on aninfinite-dimensional Teichm¨uller space is developed in [B1]. Here and in the thirdarticle (in preparation) we classify transcendental entire functions that are compo-sitions of a polynomial and the exponential for which their singular values escapeon disjoint dynamic rays. The third article is devoted to a rather special and tech-nically difficult case of such escape when the distance between different singularorbits (as between sets) is equal to zero. In the fourth article (in preparation) weprove continuity of parameter rays and their higher dimensional analogues withrespect to potentials and external addresses.1.
Introduction
An important notion in the study of dynamical behavior of transcendental entirefunctions, as in polynomial dynamics, is the escaping set . For a transcendentalentire function f its escaping set is defined as I ( f ) = { z ∈ C : f n ( z ) → ∞ as n → ∞} . It is proved in [SZ] for the exponential family { e z + κ : κ ∈ C } and in [RRRS]for more general families of functions (of bounded type and finite order) that theescaping set of every function in the family can be described via dynamic rays and heir endpoints. About the escaping points that belong to a dynamic ray we saythat they escape on rays . This is the general mode of escape for many importantfamilies of transcendental entire functions. Remark 1.1.
It is an interesting fact though, that in some cases the endpoints ofrays form a “bigger” set than the union of all rays without endpoints, in the senseof the Hausdorff dimension [Ka, S1, S2] . In this article we discuss only the escape on rays and avoid the escape on end-points because they might have a slower and less predictable orbit [Re2].We focus on entire functions of the form p ◦ exp where p is a polynomial. More-over, we assume that p is monic (otherwise replace f by its affine conjugate) anddenote by N d the set of all functions p ◦ exp where p is a monic polynomial ofdegree d . As for the exponential family [SZ], for such functions the points escapingon rays can be described by their potential (or “speed of escape”) and externaladdress (or “combinatorics of escape”, i.e. the sequence of dynamic rays containingthe escaping orbit), for details see Theorem 2.12. This is analogous to the B¨ottchercoordinates for polynomials where the points in the complement of the filled Juliaset are encoded by their potential and external angle (another more general wayto introduce “B¨ottcher coordinates” for trancendental entire functions of boundedtype is described in [Re1]).In complex dynamics it is very often important to study parameter spaces ofholomorphic functions rather than each function separately. The most famousexample of such parameter space is the Mandelbrot set. The simplest example inthe transcendental world is the space of complex parameters κ each associated tothe function e z + κ . However, for more general classes of transcendental entirefunctions, their parameter spaces have a less explicit form. Definition 1.2 (Parameter space) . Let f be a transcendental entire function. Thenthe parameter space of f is the set of transcendental entire functions ϕ ◦ f ◦ ψ where ϕ, ψ are quasiconformal self-homeomorphisms of C . For the exponential family N this definition of the parameter space coincideswith the parametrization of e z + κ (up to affine conjugation), while for d > N d is a disjoint union of different parameter spaces (again up to affine conjugation).For every function f ∈ N d , our goal is to classify those functions in the pa-rameter space of f for which all singular values escape on disjoint rays (that is,each dynamic ray contains at most one post-singular point; in particular, we avoid(pre-)periodic rays). The results in this direction for the exponential family N areproved [FRS, FS, F]. Since the complement of the Mandelbrot set in C is exactlythe set of parameters for which the critical value escapes on rays, the set we areclassifying is in a way (contained in) a transcendental analogue to the complementof the Mandelbrot set. Remark 1.3.
After certain modifications of our toolbox it is possible to relax thecondition of the disjointness of rays. Absence of this disjointness basically boilsdown to consideration of two possibilities (possibly appearing together): when twosingular points are mapped after finitely many iterations onto the same dynamicray, and when the ray containing a singular point is (pre-)periodic. The formercase is rather a notational complication: the construction in Theorem 5.9 wouldneed to take care of possible branches of rays. The latter case would require a slightlydifferent description of homotopy types “spiders” (see Subsection 4.2); in particular,one would need to introduce “spider legs with infinitely many knees” (at markedpoints), and adjust the definition of a “leg homotopy word” (Definition 4.10). Both ases would not require any essential additional analysis of the Teichm¨uller space,but would overload Theorem 5.9 with a multi-level notation. Therefore, we considerthe slightly “simpler” case of disjoint rays. For d > f ∈ N d has more than one singular orbit. If two singularpoints escape on disjoint orbits, it might happen that the (Euclidean) distancebetween them (as between sets) is equal to zero. This is a rather special case (seeTheorem 2.12) which requires serious additional analysis (see the discussion afterTheorem 5.9). Therefore, we exclude this case in this article (but fully solve it inthe next article) and prove the following theorem. Theorem 1.4 (Classification Theorem) . Let f ∈ N d be a transcendental entirefunction with singular values { v i } mi =1 . Let also { s i } mi =1 be exponentially boundedexternal addresses that are non-(pre-)periodic and non-overlapping, and { T i } mi =1 bereal numbers such that T i > t s i admitting only finitely many non-trivial clusters.Then in the parameter space of f there exists the unique entire function g = ϕ ◦ f ◦ ψ such that each of its singular values ϕ ( v i ) escapes on rays, has potential T i andexternal address s i .Conversely, every function in the parameter space of f such that its singularvalues escape on disjoint rays and with finitely many non-trivial clusters is one ofthese. The condition that external addresses are non-(pre-)periodic and non-overlappingbasically means that the singular values of g escape on disjoint rays (for details seeSection 2). That { s i } mi =1 and { T i } mi =1 admit only finitely many non-trivial clustersimplies that the distance between orbits is strictly positive (see the discussion afterDefinition 5.1).The second paragraph of the Classification Theorem 1.4 is an immediate corollaryof Theorem 2.12, and we have added it to the statement in order to show that weindeed have a classification of function in N d with singular values escaping on rays(subject to some restrictions).We are going to use machinery developed in [Bthe, B1]. It is based on generaliza-tions of the famous Thurston’s Topological Characterization of Rational Functions [DH, H] for the case of infinitely many marked points. In particular, we are goingto use the same general strategy for the proof of Classification Theorem 1.4.(1) Construct a (non-holomorphic) “model map” f for which the singular valuesescape on rays with the desired potential and external address. It will definethe Thurston’s σ -map acting on the Teichm¨uller space T f of the complementof post-singular set P f of f (Definition 3.2).(2) Construct a compact subset C f in T f which is invariant under σ .(3) Prove that σ is strictly contracting in the Teichm¨uller metric on C f .(4) Prove that σ has the unique fixed point in C f , and this point correspondsto the entire function with the desired conditions on its singular orbits.Items (1),(3) and (4) are either proved in [B1] or their proof is completely analo-gous. Therefore, the “heaviest” item is the construction of C f . The main differencefrom the exponential case in [B1] is that additional analysis of the parameter spaceis needed. It is accomplished in Section 5.1.Roughly speaking, the compact invariant subset looks the same as in the ex-ponential case. For some initially chosen constant ρ >
0, it is the set of pointsrepresented by quasiconformal maps ϕ which can be obtained from identity via anisotopy ϕ u such that:(1) for very point z ∈ P f outside of D ρ (0), ϕ u ( z ) is contained in a small disk U z around z so that the disks U z are mutually disjoint;
2) inside of D ρ (0) the distances between marked points ϕ u ( z ) are boundedfrom below;(3) the isotopy type of ϕ u relative points of P f outside of D ρ (0) is “almost”the isotopy type of identity;(4) the isotopy type of ϕ relative points of P f inside of D ρ (0) is not “toocomplicated”, i.e. there are quantitative bounds on how many times themarked points “twist” around each other.Conditions (2),(3) separate the complex plane into two subsets: D ρ (0) where wehave not so much control on the behavior of ϕ u but have finitely many markedpoints, and the complement of D ρ (0) where the homotopy information is trivial butwe have infinitely many marked points.Conditions (2)-(4) describe the position in the moduli space, while (5)-(6) encodehomotopy information of a point in the Teichm¨uller space. Structure of the article.
In Section 2 we prove that for every function in N d , thepoints escaping on rays can be encoded via their potentials and external addresses(Theorem 2.12).In Section 3 we define the Thurston’s σ -map and show that it is strictly con-tracting on the set of asymptotically conformal points in the Teichm¨uller space.Afterwards, in Section 4 we adapt the notions of id-type maps and spiders andthe associated techniques introduced in [B1] to our setting.Finally, in Section 5 we construct the invariant compact subset (Theorems 5.9and 5.11), and prove the Classification Theorem 1.4.Appendix A contains a few properties of polynomials and functions from N d thatwe occasionally use throughout the article.2. Escaping set of functions in N d In this section we prove some preliminary results about the structure of escapingset of functions in N d . Within the section we assume that d is fixed. We use theideas from [SZ] where analogous results are formulated for the exponential familyexp( z ) + κ (i.e. for N ).The first statement of the section claims that for functions in N d , as for theexponential family, escaping points escape “to the right”. Lemma 2.1 (Escape to the right) . If f ∈ N d , then f n ( z ) → ∞ if and only if Re f n ( z ) → + ∞ .Proof. | f n ( z ) | = (cid:12)(cid:12) p ◦ exp ◦ f n − ( z ) (cid:12)(cid:12) → ∞ ⇐⇒ (cid:12)(cid:12) exp ◦ f n − ( z ) (cid:12)(cid:12) → ∞ ⇐⇒ Re f n − ( z ) → + ∞ . (cid:3) Now, we introduce certain domains that help to describe combinatorics of theescaping points and play similar role as the tracts in the study of transcendentalentire functions of bounded type (in fact they are tracts of f ). By a slight abuseof terminology we also call them tracts. Lemma 2.2 (Definition and properties of tracts) . Let f ∈ N d , and let r ∈ R besuch that the right half-plane H r = { z ∈ C : Re z > r } does not contain any singularvalue of f . Then the preimage of H r under f has countably many path-connectedcomponents (called tracts ) having ∞ as a boundary point, so that for every tract T the restriction of f on T is a conformal isomorphism. oreover, for every ǫ ∈ (0 , π/ d ) we can choose r, t ∗ , t ∗ ∈ R such that (1) every tract T is contained in a right-infinite rectangular strip [ t ∗ , + ∞ ) × (cid:20) πnd − π d − ǫ, πnd + π d + ǫ (cid:21) , for some n ∈ Z , (2) every tract T contains a right-infinite rectangular strip [ t ∗ , + ∞ ) × (cid:20) πnd − π d + ǫ, πnd + π d − ǫ (cid:21) , for some n ∈ Z .Proof. Note that the restriction of f on some component T is a covering map. Butsince H r is simply-connected, its cover T must be simply-connected as well. Thismeans that f | H r is a homeomorphism, and hence a conformal isomorphism.Proof of the second part of the lemma is an elementary calculus exercise. (cid:3) From here we fix some choice of tracts, such that the conditions in the secondpart of Lemma 2.2 are satisfied. For every n ∈ Z denote by T n the tract containingthe infinite ray { t ∗ + t + 2 πin/d : t ≥ } .We are also going to make use of the following lemma. Lemma 2.3 (Exponential separation of orbits [RRRS, Lemma 3.1]) . Let f ∈ N d and T be a tract of f that is a preimage of H r . If w, z ∈ T are such that | w − z | ≥ ,then | f ( w ) − f ( z ) | ≥ exp (cid:18) | w − z | π (cid:19) (min { Re f ( w ) , Re f ( z ) } − r ) . Note that in [RRRS] the lemma is stated in a slightly different context (forlogarithmic coordinates). Nevertheless, it remains true also in our setting.In order to describe the escaping set of a function in N d we need the notions ofa ray tail and of a dynamic ray . Definition 2.4 (Ray tails) . Let f be a transcendental entire function. A ray tail of f is a continuous curve γ : [0 , ∞ ) → I ( f ) such that for every n ≥ the restric-tion f n | γ is injective with lim t →∞ f n ( γ ( t )) = ∞ , and furthermore f n ( γ ( t )) → ∞ uniformly in t as n → ∞ . Definition 2.5 (Dynamic rays, escape on rays, endpoints) . A dynamic ray of atranscendental entire function f is a maximal injective curve γ : (0 , ∞ ) → I ( f ) such that γ | [ t, ∞ ) is a ray tail for every t > .If a point z ∈ I ( f ) belongs to a dynamic ray, we say that the point escapes onrays (because in this case every iterate of the point belongs to a dynamic ray).If there exists a limit z = lim t → γ ( t ) , then we say that z is an endpoint of thedynamic ray γ . From [RRRS] we already know that the escaping set of functions in N d is orga-nized in form of the dynamic rays. Nevertheless, we reproduce the techniques from[SZ] for functions in N d , in order to obtain a more precise description of them.Let γ : [0 , ∞ ) → I ( f ) be a ray tail of f . Then for every n ≥ t n ≥ s n such that f n ◦ γ | [ t n , ∞ ) is contained in the tract T s n .Moreover, all t n except finitely many are equal to 0. This easily follows from theuniform escape on the ray tail and the fact that every connected component of thepreimage under f of small (punctured) neighborhoods of ∞ is contained in sometract. efinition 2.6 (External address) . Let z ∈ I ( f ) be a point escaping either on raysor on endpoints of rays. We say that z has external address s = ( s s s ... ) where s n ∈ Z , if either each f n ( z ) lies on a dynamic ray contained in T s n near ∞ or each f n ( z ) is the endpoint of such ray.In this case we also say that the dynamic ray either containing z or having it asendpoint has the external address s . It is clear that the external address does not depend on a particular choice of H r in the definition of tracts.On the set of all external addresses we can consider the usual shift-operator σ :( s s s ... ) ( s s s ... ). Two external addresses s and s are called overlappingif there are integers k, l ≥ σ k s = σ l s .Now we define the function which allows to characterize the “speed of escape”of points in I ( f ). We are going to refer to this function very often in different partsof the thesis. Definition 2.7 ( F ( t )) . Denote by F : R + → R + the function F ( t ) := exp( dt ) − . Note that F = F d implicitly depends on d but we suppress this from notation,this will not cause any confusion in the thesis since we do not compare functionswith different d . Remark 2.8. If d = 1 , then the function F is the same as was used for theexponential family in [SZ] . In fact, for d > our function F = F d is conjugateto F = F = exp( t ) − through an R + -homeomorphism having asymptotics t − log dd near ∞ , i.e. they are replaceable with each other. Nevertheless, for computationalreasons we prefer to use F = F d . What one should know about F is that its iterates grow very fast, as shown inthe next elementary lemma. Lemma 2.9 (Super-exponential growth of iterates) . For every t, k > we have F n ( t ) /e n k → ∞ as n → ∞ . The next definition relates two last notions.
Definition 2.10 (Exponentially bounded external address, t s ) . We say that theexternal address s = ( s s s ... ) is exponentially bounded if there exists t > suchthat s n /F n ( t ) → as n → ∞ . The infimum of such t we denote by t s . Next statement claims that no other type of external addresses can appear in I ( f ). Lemma 2.11.
Let f ∈ N d . If z ∈ I ( f ) escapes either on rays or endpoints andhas the external address s = ( s s s ... ) , then s is exponentially bounded.Proof. If Re z is big enough, then | f ( z ) | + 1 = | (1 + o (1)) exp( dz ) | < exp(Re d ( z + 1 / < F (Re z + 1) ≤ F ( | z | + 1) . Hence | Im f n ( z ) | + 1 < | f n ( z ) | + 1 < F n ( | z | + 1) . So Im f n ( z ) /F n ( t ) → t > | z | + 1 since F n ( t ) /F n ( t ) → n → ∞ for any t > t . (cid:3) ow we prove the key theorem of this section which generalizes the results of[SZ] beyond N . Theorem 2.12 (Escape on rays in N d ) . Let f ∈ N d . Then for every exponentiallybounded external address there exists a dynamic ray realizing it.If R s is dynamic ray having an exponentially bounded external address s =( s s s ... ) , and no strict forward iterate of R s contains a singular value of f , then R s is the unique dynamic ray having external address s , and it can be parametrizedby t ∈ ( t s , ∞ ) so that R s ( t ) = t + 2 πis d + O ( e − t ) , (2.1) and f n ◦ R s = R σ n s ◦ F . Asymptotic bounds O ( . ) for R σ n s ( t ) are uniform in n onevery ray tail contained in R s .If none of the singular values of f escapes, then I ( f ) is the disjoint union ofdynamic rays and their escaping endpoints.Proof. Fix some r > | f ′ ( z ) | ≥ f − ( H r ), f − ( H r ) is adisjoint union of tracts T n , n ∈ Z , each f | T n is a conformal homeomorphism, andlet L n : H r → T n be its inverse.Further, let s = ( s s s ... ) be an exponentially bounded external address. For n ≥ t ≥ t s consider the functions g n ( t ) := L s ◦ L s ◦ ... ◦ L s n − (cid:18) F n ( t ) + 2 πis n d (cid:19) , whenever they are defined.If s n /F n ( t s ) →
0, let t := t s . Otherwise fix some t > t s . Next, consider δ k ( u ) := L s k ( F ( u ) + 2 πis k +1 /d ) − ( u + 2 πis k /d ) with u ≥ F k ( t ). Since δ k ( u ) isbounded, we have f (cid:18) u + 2 πis k d + δ k ( u ) (cid:19) = F ( u ) + 2 πis k +1 d ⇐⇒ (1 + O ( e − u )) e d ( u +2 πis k /d + δ k ( u )) = (1 + O ( e − u )) e du ⇐⇒ δ k ( u ) = O ( e − u ) . The estimate above makes sense for all u ≥ F k ( t ) whenever the index k is bigenough, say bigger than some integer N ≥
0. Note that the estimate O ( . ) is uniformin k .Further, if n ≥ N , f N ◦ g n +1 ◦ F − N ( u ) − f N ◦ g n ◦ F − N ( u ) = L s N ◦ ... ◦ L s n (cid:18) F n +1 − N ( u ) + 2 πis n +1 d (cid:19) − L s N ◦ ... ◦ L s n − (cid:18) F n − N ( u ) + 2 πis n d (cid:19) = L s N ◦ ... ◦ L s n − (cid:18) F n − N ( u ) + 2 πis n d + δ n (cid:0) F n − N ( u ) (cid:1)(cid:19) − L s N ◦ ... ◦ L s n − (cid:18) F n − N ( u ) + 2 πis n d (cid:19) . Hence (cid:12)(cid:12) f N ◦ g n +1 ◦ F − N ( u ) − f N ◦ g n ◦ F − N ( u ) (cid:12)(cid:12) < C exp( − F n − N ( u ))since L s i is contracting ( | f ′ ( z ) | > T n ).Using Cauchy criterion we see that for all N big enough f N ◦ g n ◦ F − N convergesto some continuous function R σ N s ( u ) uniformly on [ F N ( t ) , ∞ ], and, moreover, R σ N s ( u ) = u + 2 πis N /d + O ( e − u ). This is a ray tail having external address σ N s .After taking N preimages of it under f (and possibly restricting it to a bigger t to void issues of the singular values), we obtain a ray tail having external address s .It is contained in a dynamic ray, so every exponentially bounded external addressis realized via some dynamic ray R s .If none of the strict forward iterates of R s contains a singular value, then, bychoosing t arbitrarily close to t s , the function can be extended to ( t s , ∞ ) (and evento [ t s , ∞ ) if s n /F n ( t s ) →
0) .The formula 2.1 for R s follows from the computations above (we can just considerbig values of u ). Since f ◦ g n +1 = g n ◦ F , we obtain the equality f n ◦ R s = R σ n s ◦ F .Injectivity of R s ( t ) follows from the asymptotic formula 2.1: points correspondingto different parameters t have distance bigger than one after iterating f some finitenumber of times.That such ray having an external address s is unique follows from strict con-traction of f − on H r : any two such rays would have had a common part near ∞ .Since they do not contain a critical value, they cannot “branch”, so must coincide.Now our goal is to show that the parametrization as above on ( t s , ∞ ) (or on[ t s , ∞ )) is the parametrization of the whole dynamic ray (possibly with the end-point).It is enough to prove that whenever two points are on the same ray R s , then thereal part of one of them grows faster than F n ( t ) for some t > t s . Indeed, becauseof the strict contraction of f − on H r , by the standard argument it would followthat this point belongs to the parametrized part of R s .Recall that in Lemma 2.11 we proved that if z ∈ I ( f ) and all iterates lie outsideof some bounded set, then | Im f n ( z ) | + 1 < F n ( | z | + 1). But this means that if z ∈ R s , then | z | + 1 ≥ t s . Analogously, for every n ≥ | f n ( z ) | + 1 ≥ t σ n s = F n ( t s ) . Taking logarithm of both sides of the inequality for n = k + 1 we obtain that forevery k ≥ f k ( z ) ≥ F k (cid:0) t s )(1 + o (1) (cid:1) where o (1) → k → ∞ .Now assume that z, w ∈ R s . Without loss of generality we might assume thatthey escape inside of the same sequence of tracts (otherwise switch to their iterates).The distance between them along the orbits has to be unbounded (otherwise theycoincide by the usual contraction argument). But then without loss of generalitywe may assume that | w − z | > (cid:12)(cid:12) f n +1 ( w ) − f n +1 ( z ) (cid:12)(cid:12) ≥ exp (cid:18) | f n ( w ) − f n ( z ) | π (cid:19) (cid:0) min { Re f n +1 ( w ) , Re f n +1 ( z ) } − r (cid:1) . Hence either z or w escapes faster than F n ( ǫ ) for some ǫ >
0, so if t s = 0, atleast one point has to be on the ray and corresponds to a parameter t > t s = 0(note that | f n ( w ) − f n ( z ) | = Re ( f n ( w ) − f n ( z )) + O (1)).Otherwise we obtain | f n ( w ) − f n ( z ) | ≥ exp (cid:12)(cid:12) f n − ( w ) − f n − ( z ) (cid:12)(cid:12) π ! (cid:0) F n ( t s )(1 + o (1)) − r (cid:1) . For big n the right hand side will grow faster than F n ( t s + ǫ ). Hence either z or w escapes faster than F n ( t s + ǫ ).Finally, we have to prove that if none of the singular values of f escapes, then I ( f ) is the union of dynamic rays and their escaping endpoints. This is a well knownfact even for more general families of functions [RRRS]. We can argue as follows. rom [RRRS, Theorem 4.7] we know that every escaping point lands after finitelymany iterations either on a dynamic ray or on an endpoint. Since rays do notcontain singular values they can be pulled back by f at their full length (togetherwith endpoints when exist), and their preimages are dynamic rays as well. Thisfinishes the proof of the theorem. (cid:3) Theorem 2.12 allows us to define the notion of potential.
Definition 2.13 (Potential) . Let f ∈ N d and z escapes on rays with externaladdress s . We say that t is the potential of z if for n → ∞| f n ( z ) − F n ( t ) − πis n /d | → . From Theorem 2.1 follows that every point escaping either on rays (withoutsingular values on forward iterates) has a potential, and different points on thesame ray have different potentials.Next lemma will help us to deal with some technicalities later on. It says that allfar enough iterates of ray tails have the strictly increasing real part. We will need itto prove that these ray tails have a “trivial” homotopy type (Proposition 5.6), evenafter some perturbations of marked points. Note that such a statement was notneeded in [B1] for the exponential family because the “triviality” of the homotopytypes was evident.
Lemma 2.14 (Ray tails are monotone near ∞ ) . Let R s = R s ( t ) be a dynamic rayof f parametrized by potential, and let t ′ be the potential of a point on the ray R s .Then there exists N > such that for all n > N the real part of f n ( R s | [ t ′ , ∞ ] ) isstrictly increasing.Proof. Recall that according to our earlier agreement f maps each tract T n confor-mally onto H r for some r ∈ R .Let R = R s ([ t ′ , + ∞ ]) and s = ( s s s ... ). For every integer n ≥ Q n := (cid:20) F n ( t ′ ) − , + ∞ (cid:19) × (cid:20) πis n d − n , πis n d + 12 n (cid:21) . First, we prove that for all n big enough holds:(1) f n ( R ) ⊂ Q n ⊂ T s n ,(2) if z ∈ Q n , then | arg f ′ ( z ) | < n − , (3) if Q − n is the connected component of f − ( Q n +1 ) containing f n ( t ′ ), then Q − n ⊂ Q n . (1) The first and the second inclusions follow immediately from the asymptoticformula 2.1 and Lemma 2.2, respectively. (2)
Note that if z = x + iy ∈ Q n , then f ′ ( z ) = ( p ◦ exp) ′ ( z ) = p ′ ( e z ) e z = de dz (cid:0) O ( e − z ) (cid:1) = de dx e idy (cid:0) O ( e − x ) (cid:1) = de dx e idy (cid:16) O ( e − F n ( t ′ ) ) (cid:17) , where O ( . ) is uniform in n .Then for big n | arg f ′ ( z ) | = (cid:12)(cid:12)(cid:12) arg y + arg (cid:16) O ( e − F n ( t ′ ) ) (cid:17)(cid:12)(cid:12)(cid:12) < n + O ( e − F n ( t ′ ) ) < n − . (3) Consider only n big enough so that: Q n ⊂ H r , • for every z ∈ f n ( R ) we have D | f ′ ( z ) | ( z ) ⊂ Q n . The second condition follows directly from the asymptotic formula 2.1 and thefact that for some constant
C > | f ′ ( z ) | < C exp( F n ( t ′ )) . If n is big enough, then from the asymptotic formula 2.1 follows that for every w ∈ Q n +1 there exists a point a ∈ f n ( R ) ⊂ Q n such that | w − f ( a ) | < . Next, since f is univalent on every tract, from Koebe 1 / f ( D | f ′ ( a ) | ( a )) ⊃ D ( f ( a )) ∋ w. Hence Q n contains a point z such that f ( z ) = w .Now we have proven the conditions (1)-(3) and are ready to finish the proof ofthe lemma.Let N > n ≥ N conditions (1)-(3) are satisfied. Take t , t such that t > t ≥ t ′ , and for n > N let γ n : [0 , → C be the straight line segmentjoining R σ n s ( t ) to R σ n s ( t ) so that γ n ( u ) = R σ n s ( t ) + u ( R σ n s ( t ) − R σ n s ( t )) . The segment γ n is evidently contained in Q n .It follows from the asymptotic formula 2.1 that we can find m ≥ N such that | arg γ m | < , in particular, Re R σ m s ( t ) > Re R σ m s ( t ).Consider the smooth curve ˜ γ ( u ) := f N − m ◦ γ m ( u ) joining R σ N s ( t ) and R σ N s ( t ).Because of (2) , for every u ∈ [0 ,
1] we havearg ˜ γ ′ ( u ) < N − + 12 N + ... + 12 m − + | arg γ m | <
12 + 12 = 1 < π . Hence Re ˜ γ is strictly increasing, and so Re R σ N s ( t ) > Re R σ N s ( t ). The samecomputations work for every n > N instead of N , i.e. Re R σ n s ( t ) > Re R σ n s ( t ). (cid:3) Setup of Thurston iteration and contraction
In this section we define the Thurston’s σ -map and discuss its properties, inparticular, that it is strictly contracting on the set of asymptotically conformalpoints in the Teichm¨uller space. The definitions are completely analogous to thosein [B1] for the exponential family, so we will be brief. See [B1] for a more detaileddiscussion.We show how to construct a quasiregular function f modeling some escapingbehavior of singular values. Let f ∈ N d , and v , ..., v m be the finite singular valuesof f . Further, let O = { a j } ∞ j =0 , ..., O m = { a mj } ∞ j =0 be some orbits of f escapingon disjoint rays R ij . Further, let R ij be the part of the ray R ij from a ij to ∞ (containing also a ij ).Now, we describe the construction of a capture map. It is a carefully chosenquasiconformal homeomorphism of C that is equal to identity outside of a boundedset and mapping singular values of f to the points { a i } . hoose some bounded Jordan domain U ⊂ C \ S i =1 ,mj =1 , ∞ R ij containing all singularvalues of f and the first point a i on each escaping orbit O i . Define an isotopy c u : C → C through quasiconformal maps, such that c = id, c u = id on C \ U , andfor each i = 1 , m we have c ( v i ) = a i . Denote c = c . Thus, c is a quasiconformalhomeomorphism mapping singular values to the first points on the orbits O i . Remark 3.1.
Note that the choice of the capture is not unique, so we just pickone of them. It will be shown in the proof of Classification Theorem 1.4 that ourresults do not depend on a particular choice of the capture.
Define a function f := c ◦ f . It is a quasiregular function whose singular orbitscoincide with {O i } mi =1 . We use the standard notation for the post-singular set of f : P f := ∪ mi =1 O i . It is also common to call P f the set of marked points .The Thurston’s σ -maps is supposed to act on T f , the Teichm¨uller space of C \ P f . Definition 3.2 (Teichm¨uller space of C \ P f ) . The
Teichm¨uller space T f of theRiemann surface C \ P f is the set of quasiconformal homeomorphisms of C \ P f modulo post-composition with an affine map and isotopy relative P f . The map σ : [ ϕ ] ∈ T f [ ˜ ϕ ] ∈ T f is defined in the standard way via Thurston’s diagram. C , P f C , ˜ ϕ ( P f ) C , P f C , ϕ ( P f ) ˜ ϕf = c ◦ f gϕ More precisely, let [ ϕ ] be a point in T f where ϕ : C → C is quasiconformal.Then due to the Measurable Riemann Mapping Theorem there exist a unique (upto postcomposition with an affine map) quasiconformal map ˜ ϕ : C → C such that g = ϕ ◦ f ◦ ˜ ϕ − is entire function. We define σ [ ϕ ] := [ ˜ ϕ ]. So σ is a continuous mapacting on T f . For more details and the proof that this setup is well defined we referthe reader to [B1]. For the classical setup we suggest [DH, H].As was mentioned in the Introduction, we are interested in fixed points of σ withexpectations that the entire map g on the right hand side of the Thurston’s diagramwould be the entire function that we need in the Classification Theorem 1.4.As for the exponential case in [B1], we will need the fact that the σ -map isstrictly contracting on the subspace of asymptotically conformal points of T f . Definition 3.3 (Asymptotically conformal points [GL]) . A point [ ϕ ] ∈ T f is called asymptotically conformal if for every ǫ > there is a compact set C ⊂ C \ P f anda representative ψ ∈ [ ϕ ] such that | µ ψ | < ǫ a.e. on ( C \ P f ) \ C . Next theorem is an immediate corollary from [B1, Lemma 4.1] and [B1, Lemma 4.3].
Theorem 3.4 ( σ is strictly contracting on as. conf. subset) . Let f = c ◦ f be thequasiregular function defined earlier in this section. Then the associated σ -map isinvariant and strictly contracting on the subset of asymptotically conformal points.
4. Id -type maps and spiders
In this section we mainly adjust the machinery developed in [B1] for the needsof our current constructions. .1. Id -type maps. Let f = c ◦ f be the quasiregular function defined in Section 3where f = p ◦ exp. We start with three definitions. Definition 4.1 (Standard spider) . S = ∪ i,j R ij is called the standard spider of f . Definition 4.2 (Id-type maps) . A quasiconformal map ϕ : C → C is of id -typeif there is an isotopy ϕ u : C → C , u ∈ [0 , such that ϕ = ϕ, ϕ = id and | ϕ u ( z ) − z | → uniformly in u as S ∋ z → ∞ . Definition 4.3 (Id-type points in T f ) . We say that [ ϕ ] ∈ T f is of id -type if [ ϕ ] contains an id -type map. As for the exponential case [B1], one can replace S in Definition 4.2 by endpointsof S , i.e. P f . It is easy to show that this would define the same subset of id-typepoints in T f . Definition 4.2 is slightly more convenient for us. Definition 4.4 (Isotopy of id-type maps) . We say that ψ u is an isotopy of id-type maps if ψ u is an isotopy through maps of id -type such that | ψ u ( z ) − z | → uniformly in u as S ∋ z → ∞ . The identity and c − are clearly of id-type, as well as a composition ϕ ◦ c of thecapture c with any id-type map.The following theorem claims that the σ -map is invariant on the subset of id-typepoints in T f in analogy with [B1, Theorem 5.5]. Theorem 4.5 (Invariance of id-type points) . If [ ϕ ] is of id -type, then σ [ ϕ ] is of id -type as well.More precisely, if ϕ is of id -type, then there is a unique id -type map ˆ ϕ such that ϕ ◦ f ◦ ˆ ϕ − is entire.Moreover, if ϕ u is an isotopy of id -type maps, then the functions ϕ u ◦ f ◦ ˆ ϕ − u have the form p u ◦ exp where p u is a monic polynomial with coefficients dependingcontinuously on u . Remark 4.6.
Conceptually the proof is the same as of [B1, Theorem 5.5] but sincewe have a more general family of functions, we have a slight upgrade on the levelof formulas.Proof.
First, since ˆ ϕ is defined up to a postcomposition with an affine map, and S contains continuous curves joining finite points to ∞ , uniqueness is obvious. Weonly have to prove existence.Since c − is also of id-type, ϕ can be joined to c − through an isotopy of id-typemaps: we can simply take the concatenation of two isotopies to identity ( ϕ ∼ idand id ∼ c − ).Let ψ u , u ∈ [0 ,
1] be this isotopy with ψ = c − and ψ = ϕ , and let ˜ ψ u be theunique isotopy of quasiconformal self-homeomorphisms of C , so that every ˜ ψ u isthe solution of the Beltrami equation ∂ z ˜ ψ u ∂ z ˜ ψ u = ∂ z ( ψ ◦ f ) ∂ z ( ψ ◦ f ) , fixing 0 and 1 (provided by the Measurable Riemann Mapping Theorem). Thenthe maps h u = ψ u ◦ f ◦ ˜ ψ − u will have the form h u ( z ) = β u q u ◦ exp( α u z ) , where α u , β u ∈ C \ { } and q u is a monic polynomial of degree d : a conformalbranched covering of C of degree d is necessarily a polynomial of the same degree,and a conformal branched covering of the punctured plane is the exponential (up to ompositions with affine maps). Moreover, α u , β u and the coefficients of q u dependcontinuously on u .Now, consider the isotopy ˆ ψ u ( z ) := α u ˜ ψ u ( z ) + log( β u ) /d , where the branch ofthe logarithm is chosen so that ψ ◦ f ◦ ˆ ψ − = f . Then g u := ψ u ◦ f ◦ ˆ ψ − u = p u ◦ expis a homotopy of entire functions so that g u ∈ N d , p = p and the polynomialcoefficients are continuous in u .Since the coefficients of p u are continuous, they are bounded on [0 , p u ( z ) = z d (1 + O (1 /z )) where O ( . ) is a bound that is uniform in u .Now, let z ∈ S , and assume that | z | is big enough (so that all subsequentcomputations are correct). Since all ψ u are of id-type, | ψ u ◦ f ( z ) − f ( z ) | < u . Let w u = w u ( z ) := exp( ˆ ψ u ( z )). Then | ψ u ◦ f ( z ) − f ( z ) | = | ψ u ◦ f ( z ) − ψ ◦ f ( z ) | = | p u ( w u ) − p ( w ) | < . Hence w du (1 + O (1 /w u )) = w d (1 + O (1 /w )) . Due to the continuity of ˆ ψ u and since ˆ ψ = id, we have w u = w (1 + o (1)) , where o ( . ) → u as z → ∞ . Henceexp( ˆ ψ u ( z )) = exp( ˆ ψ ( z ))(1 + o (1)) . Again, due to the continuity of ˆ ψ u (cid:12)(cid:12)(cid:12) ˆ ψ u ( z ) − ˆ ψ ( z ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ˆ ψ u ( z ) − z (cid:12)(cid:12)(cid:12) = o (1) . The first statement of the theorem is then proven with ˆ ϕ := ˆ ψ .The last paragraph of the theorem follows immediately from the computations. (cid:3) Theorem 4.5 implies that the notion of external address is preserved when weiterate id-type maps, in a sense that under proper normalization, i.e. when wechoose the map ˆ ϕ , the images of dynamic rays under ˆ ϕ preserve original asymptoticstraight lines. Remark 4.7.
In the sequel we keep using the hat-notation from Theorem 4.5. Thatis, ˆ ϕ denotes the unique id -type map so that ϕ ◦ f ◦ ˆ ϕ − is entire. Due to Theorem 4.5this notation makes sense whenever ϕ is of id -type. Spiders.
In this subsection we discuss infinite-legged spiders and their prop-erties. Everything is the ad hoc adaptation of the machinery developed in [B1].
Definition 4.8 (Spider) . An image S ϕ of the standard spider S under an id -typemap ϕ is called a spider . Definition 4.9 (Spider legs) . The image of a ray tail R ij under a spider map iscalled a leg (of a spider). Let V = { w n } ⊂ C be a finite set. Further, let γ : [0 , ∞ ] → ˆ C be a curve suchthat γ (0) ∈ V , γ ( ∞ ) = ∞ , γ | R + ⊂ C \ V and Re γ ( t ) → + ∞ as t → ∞ . On theset of all such pairs ( V, γ ) one can consider a map W : ( V, γ ) W ( V, γ ) ∈ F ( V ) , here F ( V ) is the free group on V . This map uniquely encodes the homotopy typeof γ (with fixed endpoints) in H r \ V where H r = { z ∈ C : Re z > r } contains V and γ (a particular value of r is irrelevant). Roughly speaking we homotope γ into a concatenation of a horizontal straight ray from γ (0) to + ∞ , and a loopwith the base point at ∞ , and by W ( V, γ ) denote the representation of the loopvia “straight horizontal” generators of the fundamental group of ( H r \ V ) ∪ ∞ . Forthe details of the construction we refer the reader to [B1, Subsection 6.1]. We aregoing to use this type of information for every leg of a spider, in order to obtain atame description of the id-type points in T f For every pair of i ∈ { , , ..., m } , j ≥ O ij := { a kl ∈ P f : l < j or l = j, k ≤ i } . Definition 4.10 (Leg homotopy word) . Let S ϕ be a spider. Then the leg homotopyword of a leg ϕ ( R ij ) is W ϕij := W ( ϕ ( O ij ) , ϕ ( R ij )) . Next theorem provides an estimate of how the leg homotopy words changes underThurston iteration.
Theorem 4.11 (Combinatorics of a preimage) . Let ϕ be of id -type. Then (cid:12)(cid:12)(cid:12) W ˆ ϕij (cid:12)(cid:12)(cid:12) < A ( j + 1) max { , (cid:12)(cid:12)(cid:12) W ϕi ( j +1) (cid:12)(cid:12)(cid:12) } , where A is a positive real number.Proof. This is a restatement of [B1, Theorem 6.16] for our context. That is, when(in the notation of [B1, Theorem 6.16]) V = ϕ ( O i ( j +1) ), γ = ϕ ( R i ( j +1) ) and W ( V, γ ) = W ϕi ( j +1) . Then if ˜ γ = ˆ ϕ ( R ij ), (cid:12)(cid:12)(cid:12) W ˆ ϕij (cid:12)(cid:12)(cid:12) < m ( j + 1) + i + 2) ( (cid:12)(cid:12)(cid:12) W ϕi ( j +1) (cid:12)(cid:12)(cid:12) + 1) < A ( j + 1) max { (cid:12)(cid:12)(cid:12) W ϕi ( j +1) (cid:12)(cid:12)(cid:12) , } , where A > (cid:3)
Finally we are ready to introduce a special equivalence relation of spiders, whichcoincides with the Teichm¨uller equivalence of the associated id-type maps.
Definition 4.12 (Projective equivalence of spiders) . We say that two spiders S ϕ and S ψ are projectively equivalent if for all pairs i, j we have ϕ ( a ij ) = ψ ( a ij ) and W ϕij = W ψij . Theorem 4.13 (Projective equivalence of spiders is Teichm¨uller equivalence) . Twospiders S ϕ and S ψ are projectively equivalent if and only if [ ϕ ] = [ ψ ] , i.e. ϕ isisotopic to ψ relative P f .Proof. If we replace the double index i, j by mj + i −
1, then we are in the settingof [B1, Theorem 6.28] (see also [B1, Remark 6.29]). (cid:3) Invariant compact subset
Preliminary constructions.
As earlier we consider the quasiconformal func-tion f = c ◦ f with m singular orbits O i = { a ij } constructed in Section 3 and theassociated σ -map. Denote also by t ij the potential of a ij and by s ij the number ofthe tract containing R ij near ∞ . The goal of this subsection is to prove a few pre-liminary lemmas that provide estimates on the pullback-spider S ˆ ϕ based differenttypes of information about a given spider S ϕ .Let P = { t i } ∞ i =1 be the set of all potentials of points in P f ordered so that t i < t i +1 . Obviously, to each potential correspond at most m points of P f , and P s eventually periodic, i.e. there exists a positive integer T ≤ m such that for all t i big enough t i + T = F ( t i ). Note that T can be strictly less than m if points fromdifferent orbits have equal potentials.Also we define a set P ′ := { ρ i : ρ i = t i + t i +1 } . It will come into action a bit later.Next definition describes a structure that might appear in the case when we havemore than one singular orbit.
Definition 5.1 (Cluster) . We say that a ij and a kl are in the same cluster Cl ( t, s ) if they have the same potential t = t ij = t kl and belong to the ray tails R ij and R kl contained in the same tract T s = T ij = T kl near ∞ .A cluster is called non-trivial if it contains more than on point of P f . The set P f is a disjoint union of clusters and each cluster contains at most m points. As can be seen from the asymptotic formula 2.1, for the points a ij and a kl with big potentials being in the same cluster is equivalent to having the samepotential and belonging to the same tract. Further, for points with big potentials,being in the same cluster implies that distance between them is very small, whereasthe distance between any pair of clusters is bounded from below.Denote by SV( f ) the singular points of f , i.e. the image of (finite) singularpoints of f under c . Since we are interested only in the parameter space of f ,without loss of generality we might assume that SV( f ) ∩ I ( f ) = ∅ , in particular,all exponentially bounded external addresses and potentials are realized. We saythat a list of external addresses { s i } mi =1 and potentials { T i } mi =1 admits only finitelymany non-trivial clusters containing more than one point, if the union of the orbitsof f realizing { s i } mi =1 and { T i } mi =1 contains finitely many non-trivial clusters. Onthe level of external addresses and potentials this means that there are only finitelymany pairs of indexes i, j and k, l such that simultaneously hold s ij = s kl and F j ( T i ) = F k ( T l ).It might be not immediately clear why in Theorem 1.4 we consider only the caseof finitely many non-trivial clusters. A short answer is that if there are infinitelymany of them, the construction of the invariant compact subset in Theorem 5.9becomes much more subtle; a particular example of what goes wrong is discussedright after it.The goal of the next lemma is to separate P f and S into two parts: one near ∞ with good estimates and “straight” dynamic rays, and one near the origin where thepoints are located somewhat more chaotically and “entanglements” of dynamic raysmight happen. This is a preparational statement solely about f and its standardspider. Lemma 5.2 (Good estimates for S near ∞ ) . There exists t ′ > such that: (1) if t i > t j > t ′ , then t i − t j > , (2) if t kl > t ij > t ′ , then | a kl | > | a ij | + 2 , (3) if ρ ∈ P ′ is bigger than t ′ , then all a ij with potential less that ρ are containedin D ρ − (0) , while all a ij ∈ P f with potential bigger than ρ are contained inthe complement of D ρ +1 (0) .Proof. Follows immediately from the asymptotic formula 2.1 and the fact that s ij / Re a ij → j → ∞ . (cid:3) For all ρ ∈ P ′ we introduce the following notation. Let D ρ := D ρ (0), and for i ∈ { , ..., m } let N i = N i ( ρ ) be the maximal j -index of the points a ij contained in ρ . For ρ > t ′ the disk D ρ contains first N i + 1 points { a i , ..., a iN i } of O i , whereasthe other points of O i are in C \ D ρ .5.1.1. Non-combinatorial estimates on ˆ ϕ ( P f ) . The following proposition concernsthe positions of points in P f under id-type maps ϕ and ˆ ϕ and under their isotopies ϕ u and ˆ ϕ u to c − . In particular, if under ϕ u points of P f outside of D ρ do notmove much, while the other points of P f stay inside of D ρ , then the first ones underˆ ϕ u do not move much as well, while those inside move inside of a left half-plane { z : Re z < ρ/ } . Proposition 5.3 (Non-combinatorial estimates on ˆ ϕ ( P f )) . There exists k > t ′ such that for all ρ ∈ P ′ ∩ [ k , ∞ ] holds the following statement.Let ϕ be an id -type map with ϕ (SV( f )) ⊂ D ρ , and let ϕ u , u ∈ [0 , be an isotopyof id -type maps such that ϕ = c − and for all u ∈ [0 , ϕ (SV( f )) ⊂ D ρ . Then: (1) ( Inside of D ρ ) If ϕ ( a i ( j +1) ) ∈ D ρ , then Re ˆ ϕ ( a ij ) < ρ . (2) ( Crossing the boundary ) If (cid:12)(cid:12) ϕ ( a i ( N i +1) ) − a i ( N i +1) (cid:12)(cid:12) < , then Re ˆ ϕ ( a iN i ) < ρ . (3) ( Outside of D ρ ) If a ij / ∈ D ρ and (cid:12)(cid:12) ϕ u ( a i ( j +1) ) − a i ( j +1) (cid:12)(cid:12) < for all u ∈ [0 , , then | ˆ ϕ u ( a ij ) − a ij | < j . Remark 5.4.
The value ρ/ in items (1)-(2) of the lemma does not have any specialmeaning. Our ultimate goal in Theorem 5.9 will be to prove that | ˆ ϕ ( a ij ) | < ρ . Sowe could have taken ερ where < ε < , but to adjust other estimates for the realand imaginary parts of ˆ ϕ ( a ij ) later on.Proof. Let g = ϕ ◦ f ◦ ˆ ϕ − = q ◦ exp with q ( z ) = z d + b d − z d − + ... + b z + b , andanalogously g u = ϕ u ◦ f ◦ ˆ ϕ − u = q u ◦ exp with q u ( z ) = z d + b ud − z d − + ... + b u z + b u and g = f .Restrict first to ρ > t ′ . (1)-(2) From Lemma 6.3 we know that if ρ is big enough and SV ( g ) ⊂ D ρ , then q − ( D r (0)) ⊂ D r (0) for r ≥ ρ .Let | w | < max i (cid:12)(cid:12) a i ( N i +1) (cid:12)(cid:12) + 1 and g ( z ) = w . Then from the previous paragraphfollows that | exp z | = (cid:12)(cid:12) q − ( w ) (cid:12)(cid:12) < max i (cid:12)(cid:12) a i ( N i +1) (cid:12)(cid:12) + 1. If F ( t ) is the potential of a i ( N i +1) for which the maximum is realized, then we haveRe z = log (cid:12)(cid:12) q − ( w ) (cid:12)(cid:12) < log | w | < log (cid:16) max i (cid:12)(cid:12) a i ( N i +1) (cid:12)(cid:12) + 1 (cid:17) < log (cid:0)(cid:0) F ( t ) + 2 π (cid:12)(cid:12) s i ( N i +1) (cid:12)(cid:12) + 1 (cid:1) + 1 (cid:1) = log (cid:0) e dt + 2 π (cid:12)(cid:12) s i ( N i +1) (cid:12)(cid:12) + 1 (cid:1) = dt + log π (cid:12)(cid:12) s i ( N i +1) (cid:12)(cid:12) + 1 F ( t ) + 1 ! . Since the external address is exponentially bounded we may assume that theabsolute value of the second summand is less that 1. At the same time ρ is chosenequal to t k + t k +1 , where t k , t k +1 are consecutive elements of P and t k +1 > t . Thismeans that t ≤ t k . If needed, increase ρ so that t k > t k /t k +1 < / d . Thus dt ≤ dt k = t k + 4 dt k − t k < t k + t k +1 − < ρ − . ence Re z < ρ/ ρ is sufficiently big.(3) First, since ϕ = c − , we have ˆ ϕ = id. Hence ˆ ϕ ( a i ( j +1) ) = a ij for all a ij ∈ P f .Consider some a ij / ∈ D ρ . We have the equality q u ◦ exp ( ˆ ϕ u ( a ij )) = ϕ u ( a i ( j +1) ) . From the asymptotic formula 2.1 we know that for ρ big enough we have | a ij | d +1 +1 ≤ (cid:12)(cid:12) a i ( j +1) (cid:12)(cid:12) , and hence ρ d +1 + 1 < (cid:12)(cid:12) a i ( j +1) (cid:12)(cid:12) . Since (cid:12)(cid:12) ϕ u ( a i ( j +1) ) − a i ( j +1) (cid:12)(cid:12) <
1, wehave ρ d +1 < (cid:12)(cid:12) ϕ u ( a i ( j +1) ) (cid:12)(cid:12) , and from Lemma 6.3 we obtain | exp( ˆ ϕ u ( a ij )) | / > ρ. Now, let ϕ u ( a i ( j +1) ) − a i ( j +1) = δ u with | δ u | <
1, or equivalently g u ( ˆ ϕ u ( a ij )) = g ( ˆ ϕ u ( a ij )) + δ u . From Lemma 6.2 we know that the coefficients of q u satisfy | b uk | < Lρ d − kd . Hence g u ( ˆ ϕ u ( a ij )) = f ( a ij ) + δ u ⇐⇒ e d ˆ ϕ u ( a ij ) (cid:18) b ud − exp( ˆ ϕ u ( a ij )) + ... + b u exp( d ˆ ϕ u ( a ij )) (cid:19) = e da ij (cid:18) b d − exp( a ij ) + ... + b exp( da ij ) + δ u exp( da ij ) (cid:19) ⇐⇒ e d ˆ ϕ u ( a ij ) O | exp( ˆ ϕ u ( a ij )) | / !! = e da ij O | exp( a ij ) | / !! . Hence, after taking logarithm of both sides, we getˆ ϕ u ( a ij ) − a ij − πin uij d = O | exp( ˆ ϕ u ( a ij )) | / ! + O | exp( a ij ) | / ! , where n uij ∈ Z . Furthermore, since | exp( ˆ ϕ u ( a ij )) | / > ρ , by increasing ρ we canmake the right hand side of the last expression arbitrarily small, in particular tomake its absolute value less than πi/d for all u . But then, since ˆ ϕ ( a ij ) = a ij , fromthe continuity of ˆ ϕ u follows that n uij = 0 for all j > N i . In particular, for all ρ bigenough holds | ˆ ϕ u ( a ij ) − a ij | < . But then O | exp( ˆ ϕ u ( a ij )) | / ! = O | exp( a ij ) | / ! andˆ ϕ u ( a ij ) − a ij = O | exp( a ij ) | / ! = O (cid:18) exp (cid:18) − Re a ij (cid:19)(cid:19) = O (cid:18) exp (cid:18) − t ij (cid:19)(cid:19) . The expression on the right tends to 0 much faster than the sequence { /j } ∞ j =1 so, after possibly increasing ρ , for a ij / ∈ D ρ we have | ˆ ϕ u ( a ij ) − a ij | < j . Note that in the proof of Proposition 5.3 (1)-(2) we have showed even a little bitmore.
Lemma 5.5 (Preimages of big disks) . Let ρ = ( t n + t n +1 ) / ∈ P ′ ∩ [ k , ∞ ] , andlet g ∈ N d be such that SV ( g ) ⊂ D ρ .Then if | g ( z ) | < max i (cid:12)(cid:12) a i ( N i +1) (cid:12)(cid:12) + 1 , we have Re z < ( d + 1) t n . No combinatorics near ∞ (almost). Proposition 5.3 is supposed to estimatethe positions of the marked points ˆ ϕ u ( P f ) when we know those of ϕ u ( P f ). Inparticular, we obtain good enough estimates for points ˆ ϕ u ( a ij ) when a ij are outsideof D ρ . Nevertheless, when a ij ∈ D ρ , we only get an upper bound for the real partsof ˆ ϕ u ( a ij ) though for the sake of invariance we want ˆ ϕ u ( a ij ) ∈ D ρ . Hence we arealso interested in bounds for imaginary part and a lower bound for the real part.Proposition 5.6 together with Theorem 4.11 will give an estimate on “how manytimes” ϕ u ( R ij ) “wraps” around ϕ u ( O ij ) whence we get bounds for imaginary partof ˆ ϕ u ( a ij ). More precisely, Proposition 5.6 says that ϕ u ( R ij ) generate boundedamount of “twists” if under ϕ u the points a ij do not move much and do not rotatearound each other.Now we want to prove that if the marked points near ∞ do not move much under ϕ u , then the corresponding images under ϕ u of ray tails R ij with big indexes havea uniformly restricted homotopy. Proposition 5.6 (No combinatorics near ∞ ) . If P f has finitely many clusterscontaining more than one point, then there exist constants k > t ′ and C > suchthat the following statement holds.If ρ ∈ P ′ ∩ [ k , ∞ ] , and ϕ u , u ∈ [0 , is an isotopy of id -type maps satisfying: (1) ϕ | C \ D ρ = id , (2) ϕ u ( a ij ) ∈ D ρ for j ≤ N i , (3) | ϕ u ( a ij ) − a ij | < /j for j > N i ,then for every u and every j > N i we have (cid:12)(cid:12) W ϕ u ij (cid:12)(cid:12) < C. Proof.
First, note that if j is big enough, than the points a ij / ∈ D ρ move under ϕ u inside of mutually disjoint disks D /j ( a ij ), and mutual distance between such disksis bounded from below by π/ d .Since for all big enough j the ray tail R ij = ϕ ( R ij ) has strictly increasingreal part (Lemma 2.14), for every u ∈ [0 ,
1] in the homotopy class (relative P f ) of ϕ u ( R ij ) there is a representative with the strictly increasing real part.Further, from the asymptotic formula 2.1 follows that there is a universal con-stant M > a ij ∈ P f , every isotopy ϕ u satisfying conditionsof the theorem, and every u ∈ [0 ,
1] there is at most M points a kl ∈ O ij withRe ϕ u ( a kl ) > Re ϕ u ( a ij ) −
1. Hence, from [B1, Lemma 6.6] and [B1, Lemma 6.8]we see that there exists a universal constant C such that for all ρ big enough andall j > N i we have (cid:12)(cid:12) W ϕ u ij (cid:12)(cid:12) < C . (cid:3) Separation of preimages in D ρ . Next definition and Proposition 5.8 are suitedto take care of two subjects. First, they help to control the distance between pointsˆ ϕ u ( a ij ) inside of D ρ . Hence this “blocks” one way of escape to ∞ in the Teichm¨ullerspace. Second, since one controls the distance between ϕ u ( a ij ) and the asymptoticvalue ϕ u ( a ), one obtains a lower bound for Re ˆ ϕ u ( a ij ). efinition 5.7 (Maximum of the derivative) . For ρ = t n + t n +1 ∈ P ′ , define M ρ := sup g ∈N d SV ( g ) ⊂ D ρ sup Re z< ( d +1) t n | g ′ ( z ) | . Note that M ρ < ∞ : from Lemma 6.2 we know that the polynomial coefficientsof g ′ are bounded. Also note that for ρ big enough and every fixed function g = p ◦ exp ∈ N d , from the representation g ′ ( z ) = p ′ ◦ exp( z ) exp( z ) of its derivative wehave sup Re z< ( d +1) t n | exp( z ) | < M ρ , and sup | z | < exp(( d +1) t n ) | p ′ ( z ) | < M ρ . Next proposition is needed to estimate mutual distance between marked pointsin D ρ . Proposition 5.8 (Separation of preimages) . Let ρ ∈ P ′ ∩ [ k , ∞ ] , and ϕ ∈ B ( ρ ) .If x, y ∈ C \ { asymptotic value of f } are such that ϕ ◦ f ( x ) , ϕ ◦ f ( y ) ∈ D max {| a i ( Ni +1) | +1 } (0) , then | ˆ ϕ ( x ) − ˆ ϕ ( y ) | ≥ | ϕ ◦ f ( x ) − ϕ ◦ f ( y ) | M ρ . Proof.
Let ρ = t n + t n +1 ∈ P ′ , and assume that a k ( N k +1) has the maximal potential t among points { a i ( N i +1) } mi =1 .Let g = ϕ ◦ f ◦ ˆ ϕ − . From Lemma 5.5 we know, that Re g − ( ϕ ◦ f ( x )) ≤ ( d + 1) t n if ϕ ◦ f ( x ) ∈ D | a k ( Nk +1) | +1 (0).Now let γ be a path joining ˆ ϕ ( x ) = g − ( ϕ ◦ f ( x )) to ˆ ϕ ( y ) = g − ( ϕ ◦ f ( y )) suchthat | γ | ≤ ( d + 1) t n . Then | ϕ ◦ f ( x ) − ϕ ◦ f ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z γ g ′ ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ M ρ | ˆ ϕ ( x ) − ˆ ϕ ( y ) | . (cid:3) Compact invariant subset.
Now we are finally ready to construct the in-variant compact subset C f . In the first theorem we present the construction andprove invariance. A statement that C f is compact will be the content of the secondtheorem afterwards. Theorem 5.9 (Invariant subset) . Let f = c ◦ exp be the quasiregular functiondefined in Section 3 so that P f contains only finitely many non-trivial clusters.Further, let ρ ∈ P ′ and C f ( ρ ) ⊂ T f be the closure of the set of points in the Te-ichm¨uller space T f represented by id -type maps ϕ for which there exists an isotopy ϕ u , u ∈ [0 , of id -type maps such that ϕ = id , ϕ = ϕ , and the following condi-tions are simultaneously satisfied. (1) (Marked points stay inside of D ρ ) If j ≤ N i , ϕ u ( a ij ) ∈ D ρ . (2) (Precise asymptotics outside of D ρ ) If j > N i , then | ϕ u ( a ij ) − a ij | < /j. (Separation inside of D ρ ) If j ≤ N i , l ≤ N k , ij = kl , and n = min { N i + 1 − j, N k + 1 − l } , then | ϕ u ( a kl ) − ϕ u ( a ij ) | > π d ( M ρ ) n . (4) (Bounded homotopy) If j ≤ N i , then (cid:12)(cid:12) W ϕ u ij (cid:12)(cid:12) < A N i +1 − j (cid:18) ( N i + 1)! j ! (cid:19) C where A and C are constants from Theorem 4.11 and Proposition 5.6, re-spectively.Then if ρ ∈ P ′ is big enough, the subset C f ( ρ ) is well-defined, invariant underthe σ -map and contains [id] . Let us give an overview of conditions (1)-(4) before the proof.Conditions (1)-(2) say that the maps ϕ have to be “uniformly of id-type”, thatis, the marked points outside of a disk D ρ have precise asymptotics, while inside of D ρ we allow some more freedom.Condition (3) tells us that the points inside of D ρ cannot come very close to eachother — it is necessary for keeping our set bounded (in the Teichm¨uller metric).Moreover, it is needed to control the distance to the asymptotic value — if a markedpoint is too close to it, then after Thurston iteration its preimage has its real partclose to −∞ , and this spoils condition (1).Condition (4) takes care of a homotopy information and provides bounds for leghomotopy words of legs with endpoints inside of D ρ . Note that analogous boundsfor marked points outside of D ρ are encoded implicitly in conditions (2) and (3). Proof of Theorem 5.9.
The proof is basically the same as in the exponential case[B1, Theorem 7.1]. Let C ◦ f ( ρ ) ⊂ C f ( ρ ) be the set of points in T f of which we takethe closure in the statement of the theorem, that is, represented by id-type maps ϕ for which there exists an isotopy ϕ u , u ∈ [0 ,
1] of id-type maps such that ϕ = id, ϕ = ϕ and the conditions (1)-(4) are simultaneously satisfied. Since the σ -map iscontinuous, it is enough to prove invariance of C ◦ f ( ρ ).First, note that for big ρ the set C ◦ f ( ρ ) contains [ c − ]: c − can be joined toidentity via the isotopy c − u where c u was constructed in Section 3, and this isotopysatisfies the conditions (1)-(4).In view of Lemma 5.2 for ρ ∈ P ′ big enough the first N i + 1 point on each orbit O i are contained in D ρ , while the other points are outside. Moreover, due to theasymptotic formula 2.1 those marked points from D ρ ∩ P f move under isotopy ϕ u only inside of D ρ , while for every a ij / ∈ D ρ the point a ij moves inside of a disk D ij of radius 1 /j , and all these disks D ρ and { D ij } are mutually disjoint and havemutual distance bigger than π/ d .For all ϕ ∈ C ◦ f ( ρ ) after concatenation with c − u we obtain the isotopy ψ u of id-typemaps with ψ = c − , ψ = ϕ and satisfying conditions (1)-(4). Note that then ˆ ψ u isan isotopy of id-type maps with ˆ ψ = id. Let g u ( z ) = ψ u ◦ f ◦ ˆ ψ − u ( z ) = p u ◦ exp( z ).We want to prove that ˆ ψ u satisfies each of the items (1)-(4): from this would followthat ˆ ϕ ∈ C ◦ f ( ρ ).We prove that each of the conditions (1)-(4) for ˆ ϕ follows from the conditions(1)-(4) for ϕ . We assume that ρ > max { k , k } . Note that since conditions of Proposition 5.6 hold, for j > N i we have (cid:12)(cid:12)(cid:12) W ψ u ij (cid:12)(cid:12)(cid:12) < C. Hence from Theorem 4.11 we get (cid:12)(cid:12)(cid:12) W ˆ ψ u ij (cid:12)(cid:12)(cid:12) < A ( j + 1) (cid:12)(cid:12)(cid:12) W ψ u i ( j +1) (cid:12)(cid:12)(cid:12) < A ( j + 1) A N i +1 − ( j +1) (cid:18) ( N i + 1)!( j + 1)! (cid:19) C = A N i +1 − j (cid:18) ( N i + 1)! j ! (cid:19) C. (1) Without loss of generality we assume that a is the asymptotic value of f (i.e. the image of the asymptotic value of f under c ). We want to obtain upperand lower bounds on the real and imaginary parts of ˆ ψ u ( a ij ) for a ij ∈ D ρ , whichwill give the desired estimate on (cid:12)(cid:12)(cid:12) ˆ ψ u ( a ij ) (cid:12)(cid:12)(cid:12) .Let j ≤ N i + 1. Then from Proposition 5.3 we know that Re ˆ ψ u ( a ij ) < ρ/ j ≤ N i , then (cid:12)(cid:12) ϕ u ( a i ( j +1) ) − ϕ u ( a ) (cid:12)(cid:12) > π d ( M ρ ) max i N i +1 , and (cid:12)(cid:12) p − u ( ψ u ( a i ( j +1) )) − p − u ( ψ u ( a )) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) exp( ˆ ψ u ( a ij )) − (cid:12)(cid:12)(cid:12) > π d ( M ρ ) max i N i +2 . Hence, Re ˆ ψ u ( a ij ) > log (cid:18) π d ( M ρ ) max i N i +2 (cid:19) . If ρ is big, the right hand side of the last expression is bigger than − ρ/
2. Thisproves that for j ≤ N i we have Re ˆ ψ u ( a ij ) < − ρ . Since (cid:12)(cid:12)(cid:12) W ψ u i ( j +1) (cid:12)(cid:12)(cid:12) < A N i − j (cid:16) ( N i +1)!( j +1)! (cid:17) C , a preimage p − u ( ψ u ( R i ( j +1) )) of the leg ψ u ( R i ( j +1) ) makes less than A N i +1 − j (cid:16) ( N i +1)! j ! (cid:17) C loops around the singular value.Hence the difference between 2 πs ij /d and the imaginary part of ˆ ψ u ( a ij ) will be lessthan 2 π (cid:18) A N i +1 − j (cid:16) ( N i +1)! j ! (cid:17) C + 1 (cid:19) . By making ρ big enough we may assumethat for all i, j such that j ≤ N i we have2 π | s ij | d + 2 π A N i +1 − j (cid:18) ( N i + 1)! j ! (cid:19) C + 1 ! < ρ . Hence, (cid:12)(cid:12)(cid:12)
Im ˆ ψ u ( a ij ) (cid:12)(cid:12)(cid:12) < ρ/ (cid:12)(cid:12)(cid:12) ˆ ψ u ( a ij ) (cid:12)(cid:12)(cid:12) = r(cid:12)(cid:12)(cid:12) Re ˆ ψ u ( a ij ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Im ˆ ψ u ( a ij ) (cid:12)(cid:12)(cid:12) < ρ/ √ < ρ . (3) Follows directly from Proposition 5.8. (2)
Follows directly from Proposition 5.3. (cid:3)
Remark 5.10.
Note that every point of C f is evidently asymptotically conformal. ow we are ready to explain why a similar construction, where marked points a ij with j > N i move under ϕ u inside of small disjoint disks around a ij , is gen-erally impossible when we have infinitely many non-trivial clusters. Considerthe function f which has three singular orbits { a ij } , i = { , , } such that thestarting points a , a , a have the same potential. Moreover, assume that a has external address s = { ... } , whereas s = { ... } and s = { ... } . Then for every j ≥
0, the point a j belongs to a clusterwith either a j or a j . For j > N + 1, if a i ( j +1) is moving under ϕ u in the diskaround a i ( j +1) with radius r i ( j +1) <
1, the point a ij is moving under ˆ ϕ u approx-imately in the disk around a ij with radius r ij = r i ( j +1) /F ′ ( t ij ) where t ij is thepotential of a ij (see computations in Proposition 5.3). Similar considerations applyto a i ( j +1) , a i ( j +2) , ... Hence, the invariance of these disks under σ would imply that r ij = 0, which is clearly impossible unless f is already entire.Next, we prove that C f ( ρ ) is compact. Theorem 5.11 (Compactness) . C f ( ρ ) is a compact subset of T f .Proof. We will prove that the set C ◦ f ( ρ ) from the proof of Theorem 5.9 is pre-compact, or equivalently, since T f is a metric space in the Teichm¨uller metric, thatevery sequence { [ ϕ n ] } ⊂ C ◦ f ( ρ ) has a subsequence that converges to a point [ ϕ ] ∈ T f .Since each [ ϕ n ] ∈ C ◦ f ( ρ ), we can assume that every ϕ n is of id-type and ϕ nu is anisotopy with ϕ n = id , ϕ n = ϕ n satisfying conditions (1)-(4) of Theorem 5.9.Recall from the proof of Theorem 5.9 that for all ρ ∈ P ′ big enough all markedpoints move under an isotopy ϕ u ∈ C ◦ f ( ρ ) inside of mutually disjoint disks D ρ and { D ij } with radii of D ij tending to 0 as j → ∞ . Hence we can assume that when u ∈ [0 , / ϕ nu | D ρ = id for every n , and when u ∈ [1 / , ϕ nu | ∪ ∞ k = N +1 D k = id for every n , that is, first the marked points inside of D ρ do notmove, and afterwards do not move the marked point outside of D ρ . It is clear fromthe definition of W ϕij (see also [B1]) that in this case for every a ij ∈ D ρ W ϕij isbounded by a constant not depending on a particular choice of ϕ . Thus, we simplyneed to prove the theorem in two separate cases: when only the marked pointsoutside of D ρ move, and when only the marked points inside of D ρ move.From this point, after relabeling of every index i, j by i + mj − (cid:3) Proof of Classification Theorem.
We are finally ready to prove the Clas-sification Theorem 1.4.
Proof of Classification Theorem 1.4.
The proof is completely analogous to the proofof [B1, Theorem 1.1].Without loss of generality we may assume that all singular values of f do not es-cape (such functions exist in the parameter space of f ). Then due to Theorem 2.12in I ( f ) there are points { a i } mi =1 so that they escape on rays with starting poten-tials { T i } mi =1 and external addresses { s i = ( s i , s i , ... ) } .Let f = c ◦ f be the captured exponential function constructed as in Section 3:i.e. with singular values { a i } mi =1 escaping as under f . From Theorems 5.9 and5.11 we know that there is a non-empty compact and invariant under σ subset C f = C f ( ρ ) ⊂ T f such that all its elements are asymptotically conformal. Next,from Theorem 3.4 follows that σ is strictly contracting in the Teichm¨uller metricon C f .Thus, we have a strictly contracting map σ on a compact complete metric space C f . It follows that C f contains a fixed point [ ϕ ]. For the details that ϕ ◦ f ◦ ˆ ϕ − s the entire required function and it is unique we send the reader to the proof of[B1, Theorem 1.1]. The proof is identical except of one irrelevant detail that wehave more than one singular orbit. (cid:3) Appendix A
In this appendix we prove some properties of polynomials and of their compo-sitions with the exponential, mainly those connecting the magnitude of the set ofsingular (or critical) values to the size of the coefficients.
Theorem 6.1 (Critical values bound critical points) . Fix some integer d > .There exists a universal constant M > such that if p is a monic polynomial ofdegree d satisfying p (0) = 0 , and the critical values of p are in the disk D ρ (0) forsome ρ > , then its critical points are in D M d √ ρ (0) .Proof. If p has all its critical points equal to 0, then p ( z ) = z d .Assume that p has all its critical values equal to 0. Then each critical point c i is a root of p of order ord( c i ) + 1 where ord( c i ) is the order of critical point (orequivalently order of the root c i of p ′ ). Since the degrees of p and p ′ differ by 1, it ispossible only if there is only one critical point c . But then p ( z ) = ( z − c ) d − ( − c ) d .Since p ( c ) = 0, it follows that c = 0. So p ( z ) = z d is the only monic polynomialsatisfying p (0) = 0 with all its critical values equal to 0.From now assume that not all critical points and not all critical values are zero.Let p ′ C ( z ) = d ( z − c ) · · · · · ( z − c d − ), where C = ( c , . . . , c d − ) ∈ C d − , and let Q ( z, C ) := p C ( z ). Then Q : C d → C is a homogeneous polynomial of degree d invariables z, c , . . . , c d − .For every C ∈ C d − \ { } let k C := max i | p C ( c i ) | max i | c i | d . From previous considerationsfollows that k C > C ∈ C d − \ { } .We want to prove that there exists a positive lower bound for k C in C d − \ { } .Then together with the degenerate case p ( z ) = z d the claim of the theorem follows.Assume that inf C d − \{ } k C = 0. Than there exists a sequence { C n } = { ( c n , . . . , c nd − ) } in C d − \ { } such that k C n →
0. Without loss of generality we can assume that | c n | = max i | c ni | > n .Consider a sequence { Γ n } := { C n /c n } = { (1 , c n /c n , . . . , c nd − /c n ) } . After chang-ing to a subsequence without loss of generality we may assume that Γ n → Γ =( γ , γ , . . . , γ d − ), where | γ i | ≤ γ = 1.On the other hand p Γ ( γ i ) = Q ( γ i , Γ) = lim n →∞ Q ( γ ni , Γ n ) = lim n →∞ Q ( c ni /c n , C n /c n ) =lim n →∞ Q ( c ni , C n ) / ( c n ) d = lim n →∞ p C n ( c ni ) / ( c n ) d That is, | p Γ ( γ i ) | ≤ lim n →∞ k C n = 0, and all critical values of p Γ are equal to 0. Aswe know, this means that p Γ ( z ) = z d , so all critical points are equal to 0 as well.But this is impossible since γ = 1. (cid:3) The following lemma provides estimates on the coefficients of polynomials p ifwe know an estimate on the magnitude of singular values of p ◦ exp. Lemma 6.2 (Singular values bound coefficients) . Fix some integer d > . Thereexists a universal constant L > such that if g = p ◦ exp where p ( z ) = z d + d − z d − + ... + b z + b , and singular values of g are contained in the disk D ρ (0) with ρ big enough, then | b k | < Lρ d − kd .Proof. If d = 1, then g ( z ) = exp( z ) + b , g has the only singular value b , and thestatement of the lemma is trivial. Hence assume that d > g in the form g = q ◦ E κ , where E κ ( z ) = exp z + κ and q ( z ) = z d + a d − z d − + ... + a z . Such representation is generally not unique so just chooseone of them. Then in particular p ( z ) = q ( z + κ ).Since g ′ ( z ) = p ′ (exp z ) exp z , the critical values of the polynomial p are thecritical values of g , possibly together with p (0) (0 is the omitted value of the ex-ponential). Hence from Theorem 6.1 we know that there is a constant M > q are contained in D M d √ ρ (0). From this, using therepresentation of the coefficients of q ′ via critical points, we conclude that there isanother constant M > | a k | < M ρ d − kd .Now we need to estimate κ . The asymptotic value of g is equal to p (0) = q ( κ )and | q ( κ ) | < ρ . We want to prove that for ρ big enough | κ | < M ρ d . Indeed,otherwise there exist arbitrarily big ρ such that | q ( κ ) | = (cid:12)(cid:12) κ d + a d − κ d − + ... + a κ (cid:12)(cid:12) = | κ | d (cid:12)(cid:12) a d − /κ + ... + a / ( κ ) d − (cid:12)(cid:12) ≥ | κ | d (1 − | a d − /κ | − ... − (cid:12)(cid:12) a / ( κ ) d − (cid:12)(cid:12) ) ≥| κ | d (1 − (cid:12)(cid:12)(cid:12)(cid:12) M ρ /d M ρ /d (cid:12)(cid:12)(cid:12)(cid:12) − ... − (cid:12)(cid:12)(cid:12)(cid:12) M ρ ( d − /d (2 M ρ /d ) d − (cid:12)(cid:12)(cid:12)(cid:12) ) > | κ | d (1 − − ... − ( 12 ) d − ) = | κ | d d − > M d ρ > ρ, which contradicts to the condition that | q ( κ ) | < ρ .Using estimates on κ , a k and Newton’s binomial formula we obtain the requiredestimate for b k . (cid:3) Next statement describes the behavior of polynomials p outside of a disk con-taining singular values of p ◦ exp subject to the condition that this disk is bigenough. Lemma 6.3 (Preimages of outer disks) . Fix some integer d > . If ρ > is bigenough then for every g = p ◦ exp with monic polynomial p of degree d and singularvalues contained in D ρ (0) we have: (1) p − ( D r (0)) ⊂ D r (0) if r ≥ ρ ; (2) p − ( D r (0)) ⊃ D ρ (0) if r ≥ ρ d +1 .Proof. Let p ( z ) = z d + b d − z d − + ... + b z + b . From Lemma 6.2 we know that ifsingular values of g are contained in D ρ (0), then | b k | < Lρ d − kd .(1) Let p ( z ) = w where | w | > ρ . Assume that | z | ≥ | w | > ρ . Then we have | w | = | p ( z ) | = (cid:12)(cid:12) z d + b d − z d − + ... + b z + b (cid:12)(cid:12) = | z | d (cid:12)(cid:12)(cid:12)(cid:12) b d − z + ... + b z d − + b z d (cid:12)(cid:12)(cid:12)(cid:12) ≥| z | d (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12) b d − z (cid:12)(cid:12)(cid:12)(cid:12) − ... − (cid:12)(cid:12)(cid:12)(cid:12) b z d − (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) b z d (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) > | w | d (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12) Lρ /d ρ (cid:12)(cid:12)(cid:12)(cid:12) − ... − (cid:12)(cid:12)(cid:12)(cid:12) Lρ ( d − /d ρ d − (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) Lρρ d (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) > | w | if ρ is sufficiently big.
2) We want to prove that p ( D ρ (0)) ⊂ D ρ d +1 (0) if ρ is sufficiently big. Let | z | < ρ . Then | p ( z ) | = (cid:12)(cid:12) z d + b d − z d − + ... + b z + b (cid:12)(cid:12) ≤ (cid:12)(cid:12) z d (cid:12)(cid:12) + (cid:12)(cid:12) b d − z d − (cid:12)(cid:12) + ... + | b z | + | b | ≤ ρ d + Lρ /d ρ d − + ... + Lρ ( d − /d ρ + Lρ < ( dL + 1) ρ d < ρ d +1 if ρ is big enough. (cid:3) Acknowledgements
We would like to express our gratitude to our research team in Aix-MarseilleUniversit´e, especially to Dierk Schleicher who supported this project from the verybeginning, Sergey Shemyakov who carefully proofread all drafts, as well as to Kos-tiantyn Drach, Mikhail Hlushchanka, Bernhard Reinke and Roman Chernov foruncountably many enjoyable and enlightening discussions of this project at differ-ent stages. We also want to thank Dzmitry Dudko for his multiple suggestions thathelped to advance the project, as well as Lasse Rempe for his long list of commentsand relevant questions. Finally, we are grateful to DFG and ERC fundings whohave financed this project.
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Parameter rays in the space of exponential maps .Ergod Theory Dynam Systems (2009), 515–544.[GL] Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichm¨uller theory . AmericanMathematical Society (2000).[H] John Hubbard,
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The dynamical fine structure of iterated cosine maps and a dimensionparadox . Duke Math J
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Email: [email protected]@univ-amu.fr