Poincar\'{e} functions with spiders' webs

For a polynomial p with a repelling fixed point w, we consider Poincar\'{e} functions of p at w, i.e. entire functions L which satisfy L(0)=w and p(L(z))=L(p'(w)*z) for all z in the complex plane. We show that if the component of the Julia set of p that contains w equals {w}, then the (fast) escaping set of L is a spider's web; in particular it is connected. More precisely, we classify all linearizers of polynomials with regards to the spider's web structure of the set of all points which escape faster than the iterates of the maximum modulus function at a sufficiently large point.


Introduction
Let f be a transcendental entire function. With the fundamental work of Eremenko [4], the escaping set I(f ) := {z ∈ C : f n (z) → ∞ as n → ∞} has become an intensively studied object in transcendental holomorphic dynamics. Since then, much progress has been achieved in exploring the topological and dynamical properties of the escaping set and some of its subsets (for some results, see [9,13,14,15,16,17]). Rippon and Stallard discovered that the fast escaping set A(f ), which was originally introduced by Bergweiler and Hinkkanen [2], shares many significant features with I(f ). If we set M(f, r) := max |z|=r |f (z)| and choose any constant R such that M(f, r) > r whenever r ≥ R, (1.1) the fast escaping set of f can be described as where A l R (f ) are the so-called level sets, defined by A l R (f ) := {z ∈ C : |f n+l (z)| ≥ M n (R), n ≥ max{0, −l}}. (Throughout the article M n denotes the n-th iterate of the maximum modulus function.) Recently, Rippon and Stallard [16,14] introduced the concept of an (infinite) spider's web. This is a connected set E ⊂ C with the property The second author has been supported by the Deutsche Forschungsgemeinschaft, Be 1508/7-1. 1 that there exists a sequence of increasing simply-connected domains (G n ) whose union is all of C such that ∂G n ⊂ E for all n. Functions whose (fast) escaping set is a spider's web have some strong dynamical properties. For instance, every such function has only bounded Fatou components and there exists no curve to ∞ on which f is bounded (compare [16]). In particular, the set of singular values of f must be unbounded. (For precise definitions see Section 2).
In [16], various sufficient criteria are presented such that I(f ) and A(f ) is a spider's web. Primarily, this is the case whenever the set A R (f ) := A 0 R (f ) is a spider's web for any R as in (1.1).
In this paper, we present a large and interesting class of functions whose escaping set is a spider's web, namely, Poincaré functions of certain polynomials. To make this precise, let p be a polynomial with a repelling fixed point z 0 (i.e. p(z 0 ) = z 0 and |p ′ (z 0 )| > 1). Then there exists an entire function L called a Poincaré function or a linearizer of p at z 0 which satisfies L(0) = z 0 and p(L(z)) = L(p ′ (z 0 ) · z) for all z ∈ C.
In the above functional equation, we can iterate the function p; this already indicates that the analysis of a linearizer strongly depends on the dynamical properties of p. However, L does not only depend on p but also on z 0 and p ′ (z 0 ) which makes linearizers good candidates for constructing functions with various interesting analytical properties (see e.g. Section 3). Furthermore, they are naturally good candidates for constructing gauge functions to estimate the Hausdorff measure of escaping and Julia sets of exponential functions (see [11]).
It was conjectured by Rempe that the escaping set of a linearizer of a quadratic polynomial for which the critical point escapes is a spider's web. In this article, we show that this is true; moreover, we classifiy all linearizers of polynomials corresponding to whether the sets A R (L) are spiders' webs or not. Theorem 1.1. Let p be a polynomial of degree d ≥ 2, let z 0 be a repelling fixed point of p and let L be a linearizer of p at z 0 . If R satisfies (1.1) then A R (L) is a spider's web if and only if the component of J (p) which contains z 0 equals {z 0 }.
Since polynomials for which all critical points converge to ∞ have totally disconnected Julia sets [5, p.85], we obtain, using [16,Theorem 1.4], the following corollary which also implies Rempe's conjecture. Corollary 1.2. Let p be a polynomial of degree d ≥ 2 for which all critical points escape and let L be a linearizer of p. Assume that R satisfies (1.1). Then each of the sets A R (L), A(L) and I(L) is a spider's web. In particular, this is true whenever p(z) = z 2 +c and c lies outside the Mandelbrot set.
We believe that the dichotomy established in Theorem 1.1 for the sets A R (L) also extends to the sets A(L) and I(L). However, we were not able to prove this. For the fast escaping set, such a result would follow if every continuum in A(f ) (or every 'loop') would be contained in some level set A l R (f ), which we also believe to be true (compare Question 2 and 3 in [16]).
In the proof of Theorem 1.1, we establish spiders' webs by proving that the corresponding linearizers grow regularly and that there exist simple closed curves arbitrary close to 0 on which the minimum modulus grows fast enough.
Since the order of a linearizer of a quadratic polynomial is given by log 2/ log p ′ (z 0 ) , we obtain for any given ρ ∈ (0, ∞) a linearizer of order ρ whose escaping set is a spider's web.

Preliminaries
The complex plane, the Riemann sphere and the unit disk are denoted by C, C := C ∪ {∞} and D, respectively. The circle at 0 with radius r will be denoted by S r . We write D r (z) for the Euclidean disk of radius r centred at z.
If not stated differently, we will assume throughout the article that f : C → C is a non-constant, non-linear entire function; so f is either a polynomial of degree ≥ 2 or a transcendental entire map.
Let C ⊂ C be a compact set. The maximum modulus M(f, C) and the minimum modulus m(f, C) of f relative to C are defined to be In the case when C = S r we will simplify the notation by writing M(f, r) and m(f, r) for M(f, S r ) and m(f, S r ), respectively. Finally, recall that the order of f is defined as If L 1 and L 2 are linearizers of f 1 and f 2 at z 1 and z 2 = ϕ −1 (z 1 ), respectively, with the same normalization, then Then L satisfies Since f 1 and f 2 are conformally conjugate, the multipliers at z 1 and z 2 coincide, hence L is a linearizer of In many dynamical settings, conformal conjugacies produce no relevant dynamical consequences, hence it is natural to ask the following: Assume that f 1 and f 2 are as in Proposition 2.1 (so f 1 and f 2 are conformally conjugate entire functions) and let L 1 be a linearizer of f 1 . Does there exist a linearizer L 2 of f 2 which is conformally conjugate to L 1 (and hence has the same dynamics)? In general, the answer is no. If namely such a linearizer L 2 would exist, then a corresponding conjugacy, say ψ, would map S(L 1 ) bijectively onto S(L 2 ), which turns out to be equivalent to the condition Since ϕ conjugates f 1 and f 2 , it already satisfies (2.2), so in particular, the map ψ −1 • ϕ is a conformal automorphism of C that fixes the set P(f 1 ). Now if Z is an arbitrary finite subset of C with at least two elements, then G Z := {h(z) = az + b : a ∈ C * , b ∈ C, h(Z) = Z} is a finite group and one can easily check that the map G Z → C * , az + b → a is an injective group-homomorphism. Hence G Z is isomorphic to a finite subgroup of C * , which must be a cyclic group generated by a root of unity. So every such G Z is generated by a map of the form z → exp(2πik/n)z + b with coprime k and n and n ≤ |Z|. This allows to phrase necessary geometric conditions on a finite set Z such that G Z is not trivial. It is clear that such conditions are rather strong; e.g., if z → exp(2πik/n)z + b is a generator of G Z and p its (unique) fixed point in C then all elements of Z must lie on r circles centred at p, where r · n ≤ |Z \ {p}|. To give an explicit dynamical example, one can consider the unique real parameter c, for which f (z) := z 2 + c has a superattracting cycle of period three; one easily sees that G P(f ) is trivial.
However, triviality of G P(f 1 ) implies ψ ≡ ϕ. So if ϕ(z) = az + b, then by Proposition 2.1, every linearizer of f 2 is of the form for some c ∈ C * , and no such map can be conformally conjugate to L 1 via ϕ whenever b = 0 (and c = 1).
Before the end of this paragraph let us observe that one can iterate f inside the functional equation and obtain as an iterated version of (2.1), where λ n denotes the function z → λ n z.
The growth of the function f and a linearizer L are related in the following sense: If f is transcendental entire then L has infinite order. If f is a polynomial then ρ(L) = log d/ log |λ|. Near ∞, the iterates of a polynomial behave in the following simple way.
Proof. The first statement is elementary and well-known. Note that we have chosen ε sufficiently small such that |z| > R ε implies |p(z)| > R ε . We will prove the statement inductivly. So for n = 1 we have q 1 (z) = 1 and the claim follows from the first part. For the iterate p n+1 (z) = p(p n (z)) we then obtain Near a repelling fixed point of p, we can make the following statement on the escaping set I(p). Proof. Let us first assume that for every δ > 0 there exists a simple closed curve γ δ ⊂ I(p) around z 0 such that dist(z 0 , γ δ ) < δ. Then J z 0 (p) is contained in the interior of every γ δ , hence it must consist of a single point.
If J z 0 (p) = {z 0 }, then for every δ > 0 there exist open, non-empty disjoint sets U δ and V δ such that J (p) ⊂ U δ ∪ V δ , J z 0 (p) ⊂ U δ and dist(z 0 , U δ ) < δ/2. Furthermore, we can assume U δ to be connected; otherwise, we replace U δ by the connected component of U δ that contains z 0 , which is also an open set. By the Plane Separation Theorem [19, Chapter VI, Theorem 3.1], there exists a simple closed curve S δ which separates z 0 from J (p) ∩ V δ such that S δ ∩ J (p) = ∅ and every point in J (p) ∩ U δ is at distance less than δ/2 from S δ . Hence, dist(S δ , z 0 ) < δ and S δ ⊂ F (p). Moreover, the component of F (p) which contains S δ must be I(p) since every bounded component of the Fatou set is simply-connected.

The set of singular values of a linearizer
If not stated differently, we will assume throughout this section that f is an entire function, z 0 a repelling fixed point of f and L a linearizer of f at z 0 . We begin with a simple connection between exceptional values of f and omitted values of L. Proof. Since L(0) = z 0 , the point z 0 is never an omitted value of L.
If a ∈ C \ E(f ), then the backward orbit of a has infinitely many elements. Since L omits at most one finite value, the backward orbit of a under f intersects L(C), i.e., there exists n ∈ N and w ∈ C with L(w) ∈ f −n (a). This means a = f n (L(w)) = L(λ n w), so a / ∈ O(L).
Now let a ∈ C \ O(L). If a = z 0 , then we are done. So suppose that a = z 0 . Then there exists z = 0 with L(z) = a. By the iterated functional equation, L(z/λ j ) ∈ f −j (a). Since z = 0 and L is injective in a neighborhood of 0, the backward orbit of a under f has infinitely many elements.
Next, we will show that the postsingular set of f and the set of singular values L coincide. This seems to be well-known (and to us, the main parts of the proof have been presented by A. Epstein), but we could not find a reference, which is why we include a proof.  Proof. Let w = L(z) ∈ C(L), in particular w / ∈ O(L). Since L ′ (0) = 0, we have w = z 0 . It follows from Proposition 3.1 that w / ∈ E(f ). Differentiating the iterated functional equation yields 0 = (f n ) ′ (L(z/λ n )) · L ′ (z/λ n ) · 1 λ n . Denote by Crit(f ) the set of critical points of f . Since L ′ (z/λ n ) = 0 if n is large enough, it follows that L(z/λ n ) ∈ Crit(f n ). Since Crit(f n ) = n−1 k=0 f k (Crit(f )) by the chain rule, there exists some k ≤ n − 1 with L(z/λ n ) = f k (y), where y ∈ Crit(f ). It follows that i.e., w ∈ n≥0 f n (C(f )).
For the other inclusion, let w ∈ f n (C(f )) \ E(f ). We want to show that there exists some z ∈ L −1 (w) with L ′ (z) = 0. Again, we differentiate the iterated functional equation and obtain There exists some y ∈ Crit(f ) such that w = f n+1 (y). Clearly, y / ∈ E(f ) since w / ∈ E(f ). By Proposition 3.1, we have y / ∈ O(L), so there exists z ∈ C with y = L(z/λ n+1 ). It follows by the chain rule that L ′ (z) = 0, and we have w = f n+1 (y) = f n+1 (L(z/λ n+1 )) = L(z), which finishes the proof of (i).
We now prove (ii). For the composition f • L one obtains since every Picard value of f is also a singular value of f . Let us abbreviate S := S(f ) ∪ f (S(L)). Since the composition L -C \ S must be a covering map as well. Hence The argument is commutative with respect to (2.1), so we obtain the opposite inclusion, yielding the equality S(L) = S(f ) ∪ f (S(L)). But for a point w ∈ S(f ), this implies that w ∈ S(L), and so f (w) ∈ f (S(L)) ⊂ S(L). By proceeding inductively, it follows for every n ∈ N that f n (w) ∈ S(L), hence P(f ) ⊂ S(L).
Let w ∈ C\P(f ). Then there exists a disk D ∋ w such that all inverse branches of all iterates of f exist in D. Let v ∈ D and z ∈ L −1 (v), and define z n := z/λ n and v n := L(z n ). Let g n be the branch of (f n ) −1 such that g n (v) = v n and let D n := g n (D). By the Shrinking Lemma in [8], it follows that the diameter of the domains D n converges to 0 (Actually, the statement in [8] is not phrased such that it completely covers our setting but the proof gives what we require). We choose a domain U in which L is injective. Then for n large enough, D n lies in L(U). Let T be the branch of L −1 that maps D n into U. Then we have Since z is an arbitrarily chosen preimage of an arbitrary point in D, all inverse branches of L can be defined in D. Hence w ∈ C \ S(L).  Proof. Let w be an attracting periodic point of f of period k and assume that w is an asymptotic value of L. Then there exists a path γ to ∞ for which lim t→∞ L(γ(t)) = w. Since w ∈ F (f ) and F (f ) is open, we can assume that L(γ) ⊂ F (f ). It follows from (2.3) that every path γ n (t) := λ −n · γ(t) is again an asymptotic path for L. Moreover, the limit of L along γ nk is contained in f −nk (w). On the other hand, every such limit point must lie in the set of attracting periodic points [3, Theorem 1], hence it follows that lim t→∞ L(γ nk (t)) = w. Furthermore, for every ε > 0 there exists N ε ∈ N such that for all n ≥ N ε , the curve L(γ nk ) intersects D ε (z 0 ). Hence z 0 ∈ ∂A * (w).
Recall that a point z ∈ J (f ) is called a buried point if it does not belong to the boundary of any Fatou component (other that I(f )). Linearizers can be very useful to construct entire or meromorphic functions whose set of singular values satisfies certain conditions. For instance, in [9], there was given an example of an entire function of finite order with no asymptotic values and only finitely many critical values such that the ramification degree on its Julia set was unbounded; the constructed function was a linearizer of a certain hyperbolic quadratic polynomial. Here we want to show another interesting example that can be constructed using linearizers, in this case of a transcendental entire function f . Let f (z) := µ exp(z) where µ ∈ C is chosen such that n≥0 f n (0) is dense in C. The existence of such parameters is well-known. By [7,Theorem 2], the function f has infinitely many fixed points. Since S(f ) = {0}, at most one of them is non-repelling [1, Theorem 7], so we can pick a repelling fixed point z 0 of f . Let L be a linearizer of f at z 0 . It follows from the functional equation that 0 is an omitted value of L. By Proposition 3.2, every point w n := f n (0) is an asymptotic value of L. It is also not hard to check that L has a direct singularity lying over each of the points w n . (For a clarification of terminology, see e.g. [3]; our last claim also follows from [3, Theorem 1.4], which is formulated for linearizers of rational maps only, but extends to linearizers of transcendental entire maps with the same proof.) Hence L is a map for which the set of projections of direct singularities (or direct asymptotic values) is dense in C. This is optimal, since by a theorem of Heins [6], the set of projections of direct singularities is always countable.

Maximum and minimum modulus estimates
In the remaining part of the article we prove Theorem 1.1. From now on, we consider an arbitrary but fixed polynomial p of degree d ≥ 2, hence p can be written as For every ε > 0 we pick a constant R ε ≥ 1 for which the conclusion of Proposition 2.2 is satisfied, and such that ε 1 < ε 2 implies R ε 1 > R ε 2 . We assume that p has a repelling fixed point z 0 with multiplier λ, and we denote by L a linearizer of p at z 0 . We also pick a constant R L ≥ 1 such that M(L, s) > s for all s ≥ R L . Lemma 4.1 (Regularity of growth). Let ε > 0, r > max{R ε , R L } and define k ε := log((1 − ε)|a d |) and K ε := log((1 + ε)|a d |). Then holds for all n ∈ N.
Proof. Let r be as assumed, and letz ∈ S r be a point for which L(z) ≥ L(z) for all z ∈ S r . Letw := L(z). Then |w| = M(L, r) and it follows from the functional equation (2.1) and Proposition 2. Moreover, we can choose R 1 = 2 · max log |a d |, log 2 |a d | , log |λ| .
Proof. Let ε ∈ (0, 1/2) be arbitrary but fixed, and let R > max{R L , R ε }. Define c R := | log kε| log R . We want to show that there exists R k such that when R > R k , m ≤ d k and n ≥ k + 1, then Obviously, it is sufficient if the wanted constant R k satisfies for all n ≥ k + 1, and this is certainly true when we choose R k sufficiently large. We will omit the details since they follow from elementary calculus; however, one can prove inductively that every R k with log R k > max{2| log k ε |, 2k d | log k ε |,  Obviously, both s and t are finite and positive constants.
Let r > |λ|·t s 1 m−1 be an arbitrary but fixed number. We define l(r) to be the unique integer for which Similarly, for the external radius t of the curve Γ δ we denote by l(t) the unique natural number for which t · |λ| l(t) ≤ r m < t · |λ| l(t)+1 .
Hence R m := max Proof of Theorem 1.1. Let us start with the case when J z 0 (p) = {z 0 }. Assume that A R (L) is a spider's web for some sufficiently large R. By definition, there exists a sequence of bounded simply-connected domains G n such that G n ⊂ G n+1 , ∂G n ⊂ A R (L) for n ∈ N, and G n = C. We can assume w.l.o.g. that every G n contains 0 (since this is true anyway for all sufficiently large n).
By Proposition 2.3, for every n ∈ N, the curve L(∂G n ) intersects the filled Julia set of p. Let K > 0 be the radius of the smallest disk around 0 which contains the (filled) Julia set of p. Then there exists a sequence of points w n ∈ ∂G n such that |L(w n )| ≤ K. But this contradicts the assumption that all points z ∈ ∂G n satisfy |L(z)| ≥ M(L, R).
Let us now consider the situation when J z 0 (p) = {z 0 }. By [16,Theorem 8.1] it is sufficient to find a sequence of bounded simplyconnected domains G n such that for all (sufficiently large) n, G n ⊃ {z ∈ C : |z| < M n (L, R)} and G n+1 is contained in a bounded component of C \ L(∂G n ).
Let R 1 be the constant from Lemma 4.2, and set R := max{R L , R 1 }. For n ∈ N let r n := |λ| n M n (L, R) (see also Lemma 4.2). By Lemma 4.3, there exists a simple closed curve Γ rn separating S rn and S r d n such that m(L, Γ rn ) > M(L, r n ). We define G n to be the interior of Γ rn . Then every G n is a bounded simply-connected domain with G n ⊃ {z ∈ C : |z| < r n } ⊃ {z ∈ C : |z| < M n (L, R)}. hence G n+1 is contained in a bounded component of C \ L(∂G n ) and the claim follows.
Note that Corollary 1.2 is an immediate consequence of Theorem 1.1.