On Poincaré Series of Unicritical Polynomials at the Critical Point
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Abstract
In this paper, we show that for a unicritical polynomial having a priori bounds, the unique conformal measure of minimal exponent has no atom at the critical point. For a conformal measure of higher exponent, we give a necessary and sufficient condition for the critical point to be an atom, in terms of the growth rate of the derivatives at the critical value.
Keywords
Complex dynamics Julia sets Poincaré series Summability conditionMathematics Subject Classification (2010)
37F351 Introduction
In this paper, we shall study atoms of conformal measures of polynomials having precisely 1 critical point; we call such a polynomial unicritical. Note that if f is a unicritical polynomial, then its degree d is at least 2, and there is c in \(\mathbb {C}\) such that f is affine conjugate to the polynomial z ^{ d }+c. A unicritical polynomial written in this form is normalized. We shall make the following assumption.
Definition 1
Let f be a unicritical polynomial whose critical point is nonperiodic and recurrent. Assume for simplicity that f is normalized, so its critical point is 0. Then we say f has a priori bounds, if there exists τ>0 such that for each ε>0 there exists a topological disk V containing 0, satisfying \(\operatorname{diam}(V)<\varepsilon \), and such that the following holds: For each integer n≥1 such that f ^{ n }(0) is in V, the connected component U of f ^{−n }(V) that contains 0 satisfies \(\overline{U} \subset V\), and there is annulus A contained in \(V \setminus\overline{U}\), enclosing U, and whose modulus is at least τ.

f ^{ s }:U→V is dto1;

U contains the critical point of f, and for each j in {1,…,s−1} the set f ^{ j }(U) does not contain it;

The set {z∈U:f ^{ sn }(z)∈U,n=0,1,…} is a connected proper subset of J(f).
In the statement of the following theorem we use the fact that for a unicritical polynomial f having a priori bounds there is a unique conformal measure of exponent t=HD_{hyp}(f) for f (see [16, Theorem 1]).
Theorem A
Let f be a unicritical polynomial having a priori bounds. Then the conformal measure μ of minimal exponent of f does not have an atom at the critical point of f.
We remark that in this result, it is essential that we consider the conformal measure of minimal exponent, as opposed to a conformal measure of higher exponent. In fact, every unicritical map f that is not uniformly hyperbolic has, for each t>HD_{hyp}(f), a conformal measure of exponent t (see [3]) for the case f satisfies the ColletEckmann condition, and [18, 19] for the case f does not. In many cases, even if f has a priori bounds, for each t>HD_{hyp}(f) there is conformal measure of exponent t that is supported on the backward orbit of the critical point.

If a conformal measure of exponent t has an atom at w, then \(\mathcal {P}(w, t)<\infty\);

A conformal measure of exponent t has an atom at the critical point c _{0} if and only if \(\mathcal {P}(c_{0}, t)<\infty\).
Theorem B
Let f be a unicritical polynomial having a priori bounds, and let t>HD_{hyp}(f) be given. Assume f is normalized so its critical point is 0. Then the series \(\mathcal {P}(0, t)\) is finite if and only if \(\sum_{n=0}^{\infty}Df^{n}(f(0))^{t/d}\) is finite.
1.1 Strategy and Organization
We derive Theorem A from Theorem B arguing by contradiction. If the conformal measure of minimal exponent had an atom at the critical point, by Theorem B the derivatives at the critical value would grow to infinity. Together with the a priori bounds hypothesis, this implies that the map is “backward contracting” (Theorem 1), so the results of [24] apply to this map; in particular, the Poincaré series at the critical point diverges. This contradicts the existence of an atom at the critical point. In Sect. 2 we fix some notation and terminology and recall the definition of backward contracting maps. We deduce Theorem A from Theorem B in Sect. 3, after proving Theorem 1.
To prove Theorem B, we show that in either case the map is backward contracting Theorem 1(1). This allows us to use the characterization for backward contracting maps of the summability condition given in [9]. We recall this result in Sect. 3, as Theorem 1(2). To prove the direct implication in Theorem B, we divide the integral of the backward contraction function, characterizing the summability condition, into integrals over intervals bounded by consecutive close return scales. Then we show that each of these integrals is bounded by one of the terms in the Poincaré series up to a multiplicative constant (Lemma 5). This is done in Sect. 4. The proof of the reverse implication in Theorem B occupies Sect. 5 and is more involved. We use a discretized sequence of scales to code each iterated preimage of the critical point. To do this, we consider the largest scale whose pullback is conformal, and consider the “critical hits” when pulling back the previous scale, as a code. One of the crucial estimates is the contribution in the Poincaré series of those iterated preimages of the critical point for which a certain ball can be pulled back conformally (Lemma 10). This estimate is done using one of the main results of [24]. For a backward contracting map the diameter of pullbacks decreases at least at a superpolynomial rate.
2 Preliminaries
We endow \(\mathbb {C}\) with its norm distance, and for a bounded subset W of \(\mathbb {C}\) we denote by \(\operatorname{diam}(W)\) the diameter of W. Moreover, for δ>0 and for a point z of \(\mathbb {C}\), we denote by B(z,δ) the ball of \(\mathbb {C}\) centered at z and of radius δ.
A topological disk is an open subset of \(\mathbb {C}\) homeomorphic to the unit disk, and that is not equal to \(\mathbb {C}\). We endow such a set with its hyperbolic metric. If V and V′ are topological disks such that \(\overline{V'} \subset V\), we denote by \(\operatorname{mod}(V; V')\) the supremum of the moduli of all annuli contained in V∖V′ that enclose V′ (see [1, 15] for background).
Throughout the rest of this paper we fix an integer d≥2, a parameter c in \(\mathbb {C}\), and put f(z):=z ^{ d }+c. Moreover, we assume 0 is nonperiodic and recurrent by f; this implies that 0 is in the Julia set of J(f). Given a subset V of \(\mathbb {C}\), and an integer n≥0, each connected component of f ^{−n }(V) is called a pullback of V by f ^{ n }. When n≥1, a pullback of V is critical if it contains 0.
2.1 Backward Contraction
In this section we recall the definition of “backward contraction” from [23], and compare it with a variant from [9].
Lemma 1
There is a constant ρ _{0}>1 independent of f and d, such that for every δ>0 satisfying \(R(\delta) \ge\rho_{0}^{d}\), and every δ′ in [δ/R(δ),δ), we have \(r(\delta') \ge\rho_{0}^{d} \delta/ \delta'\). In particular, if there is \(R_{0} \ge\rho_{0}^{d}\) such that for every sufficiently small δ>0 we have R(δ)≥R _{0}, then f is backward contracting with constant \(\rho_{0}^{d} R_{0}\). Moreover, f is backward contracting if and only if R(δ)→∞ as δ→∞.
Proof
2.2 Nice Sets and Children
The purpose of this section is to prove a general lemma about backward contracting maps that is used in Sect. 5.1 to prove Theorem B.
For an integer m≥1, and a connected open set V, a pullback W of V by f ^{ m } containing 0 is a child of V, if f ^{ m } maps W onto V as a dto1 map.
The following lemma is a more precise version of [24, Lemma 3.15], with the same proof.
Lemma 2
Proof
For each integer k≥1, let Y _{ k } be the kth smallest child of V and let s _{ k } be the integer such that \(f^{s_{k}}(Y_{k}) = V\). By the backward contracting property, we have \(\operatorname{diam}(f(Y_{1})) \le R(\delta)^{1}\delta\). Note that for each k≥1 the set \(f^{s_{k}}(Y_{k+1})\) is contained in a return domain of V, so \(\operatorname{mod}(V; f^{s_{k}}(Y_{k+1})) \ge\lambda\). By the definition of child, the map \(f^{s_{k}1} : f(Y_{k}) \to V\) is conformal, thus \(\operatorname{mod}(f(Y_{k}); f(Y_{k+1}))=\operatorname{mod}(V;f^{s_{k}}(Y_{k+1})) \ge\lambda\) and therefore \(\operatorname{diam}(f(Y_{k + 1})) \le \operatorname{diam}(f(Y_{k}))/2\). The conclusion of the lemma follows. □
3 A Priori Bounds and Backward Contraction
In this section we derive Theorem A from Theorem B. To do so, we first establish a sufficient criterion for a unicritical map having a priori bounds to be backward contracting (Theorem 1 below).
In the following theorem we summarize and complement results in [8, 9], when restricted to unicritical maps. We state it in a stronger form than what is needed for this section.
Theorem 1
 (1)Let R _{0}>1 be such that for every sufficiently large n we have eitherThen for every sufficiently small δ>0 we have R(δ)≥R _{0}.$$\biglDf^n(c)\bigr \ge\eta R_0, \quad \text{\textit{or}}\quad \min_{\zeta\in f^{n}(0)}\biglDf^n(\zeta)\bigr \ge(\eta R_0)^{1/d}. $$
 (2)
For each t>0, the sum \(\sum_{n = 0}^{\infty} Df^{n}(c)^{ t}\) is finite if and only if \(\int_{0+}^{1} R(\delta)^{t} \ \frac{d\delta }{\delta}\) is finite.
When Df ^{ n }(c) is eventually bounded from below by ηR _{0}, part 1 is given by (the proof of) [8, Theorem A]. Part 2 is a direct consequence of part 1 and [9, Theorems 1.3 and 1.4], together with Lemma 1 and [23, Corollary 8.3].
To prove this theorem, we shall use the following variation of the Koebe distortion theorem.
Lemma 3
Proof
Lemma 4
Proof
Let C _{@} be the constant given by Lemma 3. Let n≥1 be an integer such that f ^{ n }(0) is in V′, and let U (resp. U′) be the pullback of V (resp. V′) by f ^{ n } containing 0. It suffices to consider the case that U is not contained in any critical pullback of V′. In this case we claim that f ^{ n }:U→V is dto1. Otherwise, there would exist m in {1,…,n−1} such that f ^{ m }(U) contains 0. This implies that f ^{ m }(U) is contained in the pullback of V by f ^{ n−m } containing 0, which is contained in V′. Thus f ^{ m }(U) is contained in V′, and therefore U is contained in the pullback of V′ by f ^{ m } containing 0. We thus obtain a contradiction that shows that f ^{ n }:U→V is dto1. Then the lemma follows from Lemma 3 with C _{!}=C _{@}. □
Proof of Theorem 1
As mentioned above, part 2 is a direct consequence of part 1, and the combination of [9, Theorems 1.3 and 1.4], Lemma 1, and [23, Corollary 8.3].
To prove part 1, note that for δ>0 the number \(\operatorname{mod}(\widetilde {B}(\delta); \widetilde {B}(\delta/2))\) is independent of δ; denote it by τ _{1}. On the other hand, let τ be the constant given by the a priori bounds hypothesis. Note that there is a constant Δ>0 such that if U is a topological disk satisfying \(\overline{U} \subset V\) and \(\operatorname{mod}(V; U) \ge\tau\), then the diameter of U with respect to the hyperbolic metric of V is bounded by Δ. Moreover, the number \(\operatorname{mod}( V; B_{V}(0, \Delta))\) is bounded from below by a constant τ _{0}>0 that is independent of V. Let C _{!} be the constant given by Lemma 4 with τ _{0} replaced by min{τ _{0},τ _{1}}. Let ξ>1 be the constant defined above for this choice of Δ, and put η:=4C _{!} ξ ^{ d }.
Proof of Theorem A assuming Theorem B
Let μ denote the conformal measure of f of minimal exponent h=HD_{hyp}(f). Assume by contraction that μ has an atom at 0. Then \(\mathcal {P}(0, h)<\infty\), hence for each t>h we have \(\mathcal {P}(0, t)<\infty\). By Theorem B, it follows that \(\sum_{n=0}^{\infty}Df^{n}(c)^{t/d}<\infty\), hence Df ^{ n }(c)→∞ as n→∞. By Lemma 1 and part 1 of Theorem 1, the map f is backward contracting. But then, [24, Proposition 7.3] implies \(\mathcal {P}(0, h)=\infty\). We thus obtain a contradiction that completes the proof of the theorem. □
4 Convergence of Poincaré Series Implies Forward Summability
The purpose of this section is to prove the following proposition, giving one of the implications in Theorem B.
Proposition 1
Suppose f satisfies the a priori bounds condition. Then for each t>0 such that \(\mathcal {P}(0, t)\) is finite, the sum \(\sum_{n=1}^{\infty}Df^{n}(c)^{t/d}\) is also finite.
For each δ>0, let n(δ) be the minimal integer n≥1 such that f ^{ n }(0) is in \(\widetilde {B}(\delta)\), let U _{ δ } denote the pullback of \(\widetilde {B}(\delta)\) by f ^{ n(δ)} that contains 0, and let ζ(δ) be a point of f ^{−n(δ)}(0) in U _{ δ }. Clearly, n(δ) is nonincreasing with δ, left continuous, and we have n(δ)→∞ as δ→0. In view of part 2 of Theorem 1, Proposition 1 is a direct consequence of the following lemma.
Lemma 5
The proof of this lemma is after the following one.
Lemma 6
Assume that f is backward contracting, and for each δ>0 let W _{ δ } be the pullback of \(\widetilde {B}(2\delta)\) by f ^{ n(δ)−1} containing c. Then for every sufficiently small δ we have \(R(\delta) \ge \delta/ \operatorname{diam}(W_{\delta})\).
Proof
Let δ _{0}>0 be sufficiently small so that for every δ in (0,δ _{0}) we have r(δ)>2. It suffices to show that for each δ in (0,δ _{0}), and every integer m≥0 such that f ^{ m }(c) is in \(\widetilde {B}(\delta)\), the pullback W of \(\widetilde {B}(\delta)\) by f ^{ m } containing c is contained in W _{ δ }. Clearly m≥n(δ)−1, and when m=n(δ)−1 we have W⊂W _{ δ }. If m≥n(δ), then our hypothesis r(δ)>2 implies that f ^{ n(δ)−1}(W) is contained in \(\widetilde {B}(2 \delta)\). This shows that W is contained in W _{ δ }, as wanted. □
Proof of Lemma 5
By Koebe Distortion Theorem there is a constant K>1 such that for every δ>0, every integer n≥1, and every pullback W of \(\widetilde {B}(2\delta)\) by f ^{ n } for which the corresponding pullback of \(\widetilde {B}(4 \delta)\) is conformal, the distortion of f ^{ n } on W is bounded by K. Let δ _{0}>0 be such that the conclusion of Lemma 6 holds for every δ in (0,δ _{0}). Reducing δ _{0} if necessary, assume that for each δ in (0,4δ _{0}) we have R(4δ)≥4.
5 Forward Summability Implies Backward Summability
In this section we complete the proof of Theorem B. After some estimates in Sect. 5.1, the proof of Theorem B is given in Sect. 5.2.
5.1 Thickened Grandchildren
The purpose of this section is to prove the following.
Proposition 2
The proof of this proposition is given after the following lemma.
Lemma 7
Proof
In view of Lemma 7, Proposition 2 is a direct consequence of the following lemma.
Lemma 8
Proof
5.2 Proof of Theorem B
The proof of Theorem B is at the end of this section, after a couple of lemmas.
Assume f is backward contracting, fix t>HD_{hyp}(J(f)), and consider the notation introduced in Sect. 5.1 for this choice of t. Put \(\mathcal {O}^{}(0): = \bigcup_{m=1}^{\infty}f^{m}(0)\), and for each z in this set denote by m(z)≥1 the unique integer m≥1 such that f ^{ m }(z)=0. Note that for every z in \(\mathcal {O}^{}(0)\) there is an integer q≥0 such that the pullback of V _{ q } by f ^{ m(z)} containing z is conformal. Denote by q(z) the least integer q≥0 with this property.
Lemma 9

m(z)≥n, and ζ(z):=f ^{ m(z)−n}(z) is in Z _{ n }(q−1), and hence in \(V_{p_{\mathbf {n}}(q  1) + 1}\);

f ^{ m(z)−n} maps a neighborhood U(z) of z conformally onto \(V_{p_{\mathbf {n}}(q  1)}\);
 Denoting by ζ′(z) the unique point in U(z) such that f ^{ m(z)−n}(ζ′(z))=0, we have$$\biglDf^{m(z)}(z)\bigr \ge C_{\&} \bigl \vert Df^{\mathbf {n}} \bigl(\zeta(z)\bigr) \bigr \vert \bigl \vert Df^{m(z)  \mathbf {n}}\bigl( \zeta'(z)\bigr) \bigr \vert . $$
Proof
The third assertion follows from the first and the second, together with Koebe distortion theorem. To prove the first and second assertions, we proceed by induction in m(z). Let z be a point in \(\mathcal {O}^{}(0)\) such that q(z)≥1 and m(z)=1. Then 1 is in \(\mathcal {N}(q  1)\), and the desired assertions are easily seen to be satisfied with n=(1). Let m≥2 be an integer and suppose the desired assertions are satisfied for every z in \(\mathcal {O}^{}(0)\) such that q(z)≥1 and m(z)≤m−1. Let z be a point in \(\mathcal {O}^{}(0)\) such that q:=q(z)≥1 and m(z)=m. Note that f(z) is in \(\mathcal {O}^{}(0)\) and m(f(z))=m(z)−1. If q(f(z))≤q−1, then the pullback of V _{ q−1} by f ^{ m(z)−1} containing f(z) is conformal. This implies that m(z) is in \(\mathcal {N}(q  1)\), and then the desired assertions are verified with n=(m(z)). If q(f(z))≥q, then we can apply the induction hypothesis with z replaced by f(z); let \(\mathbf {n}' = (n_{1}', \ldots, n_{k'}')\) be the corresponding element of \(\mathcal {S}(q  1)\). If the pullback of U(f(z)) by f containing z is conformal, then the desired assertions are verified with n=n′. Otherwise, n′:=m(z)−n′ is in \(\mathcal {N}(p_{\mathbf {n}'}(q  1))\), and then the desired assertions are verified with \(\mathbf {n}= (n_{1}', \ldots, n_{k'}', n')\). □
Lemma 10
Proof
Proof of Theorem B
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