REAL ANALYTICITY OF TOPOLOGICAL PRESSURE FOR PARABOLICALLY SEMIHYPERBOLIC GENERALIZED POLYNOMIAL-LIKE MAPS

BERND O. STRATMANN AND MARIUSZ URBAŃSKI

Abstract. For arbitrary parabolically semihyperbolic generalized polynomial-like maps f, we prove that on a certain interval, which contains the interval (0, HD(J(f))), the pressure function t ↦ P(−t log |f′|) is real-analytic. Our results generalize the work of Makarov and Smirnov in [3] and [7].

1. Introduction and Statement of Results

In this paper we consider parabolically semihyperbolic generalized polynomial-like maps f. We show that on a certain interval which contains the interval (0, HD(J(f)), the associated topological pressure P(−t log |f′|) is real-analytic as a function in t. Here, HD(J(f)) refers to the Hausdorff dimension of the corresponding Julia set J(f). Roughly speaking, we obtain these results by showing how to associate to f some finitely primitive conformal graph directed Markov system. This then allows to use a result of [5], which states that for this type of Markov system the pressure function is real-analytic in the relevant range. We remark that our paper extends results by Makarov and Smirnov obtained in [3] [7]. Also, note that our method of associating to f a graph directed Markov system is completely different from the method used in [3] and [7]. That is, we do not have to use the construction of Hofbauer towers, nor do we have to introduce a ‘new Riemann metric’ in order to force some proper expansion.

Before we state our main result more precisely, we introduce some preliminary concepts and notation. Let U ⊂ C I be an open Jordan domain with smooth boundary, and let U := ⋃_{i∈I} U_{i} be a finite union of Jordan domains which are fully contained in U and which have pairwise disjoint closures. A generalized polynomial-like mapping (abbreviated as GPL-map) is a map

f : U → U

which has a holomorphic extension to an open neighbourhood of U such that for each i ∈ I the restriction of this extension to U_{i} is a surjective branched covering map. The set of parabolic periodic points of f is defined by

Ω := {ω ∈ U : f^{q}(ω) = ω and (f^{q})′(ω) = 1 for some q ≥ 1}.

Without loss of generality, we shall assume that all parabolic periodic points of f are in fact fixed points of f, and that f′(ω) = 1 for each ω ∈ Ω (this is of course achieved by taking a suitable iterate of f, which does not affect our analysis here, since P(−t log |f′|) = \frac{1}{n}P(−t log |(f^{n})′|), for each n ∈ ℕ). Also, we define

Research partially supported by the NSF Grant DMS 0100078.

Crit(f) := {c : f′(c) = 0} and Crit(J(f)) := J(f) ∩ Crit(f).

It will be convenient to split up the index set I in the following way.

• I_{o} := \{i ∈ I : \overline{U_{j}} ∩ ⋃_{n≥1} f^{n}(Crit(f)) = ∅\} (‘post-critical free’),

• I_{p} := \{i ∈ I : Ω ∩ \overline{U_{i}} ≠ ∅\} (‘parabolic’),

• I_{c} := {i ∈ I : U_{i} ∩ Crit(f) ≠ ∅} (‘critical’),

• I_{r} := I \ (I_{o} ∪ I_{p}) (‘regular’). Furthermore, we define

U_{o} := ⋃_{i∈I_{o}} U_{i}, U_{p} := ⋃_{i∈I_{p}} U_{i}, U_{c} := ⋃_{i∈I_{c}} U_{i}, U_{r} := ⋃_{i∈I_{r}} U_{i}.

Definition. A GPL-map f is called parabolically semihyperbolic if and only if the following conditions are satisfied.

(a) I_{c} ⊂ I_{o}, (b) \overline{U_{o} ∪ U_{r}} ⊂ U, (c) ⋃_{n≥1} f^{n}(Crit(f)) ⊂ U_{r}.

Note that in this definition we do not rule out the possibility that Ω = ∅. That is, we let the class of semihyperbolic GPL-maps be contained in the class of parabolically semihyperbolic GPL-maps. Also, recall that a GPL-map f is called non-recurrent if for each c ∈ Crit(J(f)) we have that U_{i} ∩ {f^{n}(c) : n ≥ 1} = ∅, where i ∈ I is uniquely determined by the fact that c ∈ U_{i}. Hence, by (a) in the definition above, a parabolically semihyperbolic GPL-map is always non-recurrent. In fact even more can be said, namely that such a GPL-map is parabolically subhyperbolic and critically tame (see [8] for the definitions). Furthermore, we remark that for a parabolically semihyperbolic GPL-map the sets I_{o}, I_{p} and I_{r} are always pairwise disjoint. Throughout, we shall assume that if i ∈ I_{p}, then the map f : U_{i} → U is a conformal homeomorphism. It then follows from Schwarz’s lemma that Ω ∩ \overline{U_{i}} is a singleton, denoted by ω_{i}, and that ω_{i} is contained in the boundary of U_{i}. Also, let f_{i}^{−1} : \overline{U} → \overline{U_{i}} refer to the inverse branch of f for which f_{i}^{−1}(ω_{i}) = ω_{i}. By the Denjoy-Wolf theorem, it follows that f_{i}^{−n}(z) converges to ω_{i} uniformly, for z ∈ U. Since f_{i}^{−1} has an analytic extension to an open neighbourhood of ω_{i} and since (f_{i}^{−1})′(ω_{i}) = 1, there exists a Taylor expansion of this extension, which for z close to ω_{i}, and for some fixed a_{i} ≠ 0 and p_{i} ∈ ℕ, has the form

f_{i}^{−1}(z) = z + a_{i}(z − ω_{i})^{p_{i}+1} + ….

Using this, we obtain that for each compact set F ⊂ U there exists a constant C_{F} ≥ 1 such that for every n ≥ 1 and for all z ∈ F we have that (see e.g [1])

C_{F}^{−1}n^{− \frac{p_{i}+1}{p_{i}}} ≤ |(f_{i}^{−n})′(z)| ≤ C_{F} n^{− \frac{p_{i}+1}{p_{i}}} . (LBP)

Furthermore, the following ‘critical parameters’ will be crucial in our analysis. Recall that for c ∈ Crit(J(f)), the order q(c) of c is determined by the local behaviour of f around c. That is, for z sufficiently close to c we have for f the Taylor expansion

f(z) = f(c) + b_{0}(z − c)^{q(c)} + …. (LBC)

These ‘critical parameters’ are the following.

χ_{0} := inf\{lim sup_{n→∞} \frac{1}{n} log |(f^{n})′(z)| : z ∈ Ω ∪ ω(Crit(f))\},

χ(c) := lim inf_{k→∞} \frac{1}{k} log inf_{n≥1}\{|(f^{k})′(f^{n}(c))|\},

χ_{q} := min \{\frac{χ(c)}{q(c)} : c ∈ Crit(f)\} and χ := min\{χ_{q}, χ_{0}\}.

For an introduction and discussion of the topological pressure function P for GPL-maps the reader is referred to Section 2. We can now state the main results of this paper as follows.

Theorem 1.1. Let f be a parabolically semihyperbolic GPL-map. For each positive t for which P(t) > −χt we have that the topological pressure function P is real-analytic.

Note that if f has parabolic elements, then χ_{0} = χ = 0, and P(t) = 0 for all t ≥ HD(J(f)) (see section 2). Hence, in this case the theorem exactly states that P is real-analytic in the interval (0, HD(J(f))). However, if f has no parabolic elements, then the parameters t satisfying the assumptions of the theorem may also be larger than HD(J(f)). Therefore, we have the following immediate corollary.

Corollary 1.2. For a parabolically semihyperbolic GPL-map f, the associated topological pressure function is real-analytic in the interval (0, HD(J(f))).

We remark that these results generalize the results in [3] and [7] mainly in two ways. First, our results do not require that there is exactly only one critical point contained in the Julia set. Secondly, we also allow the presence of parabolic periodic points. Another important difference in comparision with [3] and [7] is that we use a completely different method, which is based primarily on the progress on graph directed Markov systems obtained in [2], [5] and [9].

Acknowledgement: We should like to thank the Mathematics Department at the University of Bourgundy in Dijon for its warm hospitality and excellent working conditions while discussing the material of this paper. The second author extends these thanks to the Technical University of Clausthal where part of the research on this article has been conducted.

2. Review of topological pressure and conformal Gibbs states

Let us first recall the classical definition of pressure, and formulate the variational principle. Let T : X → X be a continuous map of a compact metric space (X, d) into itself. For x, y ∈ X and n ≥ 0, the metric d_{n} is defined by

d_{n}(x, y) := max{d(T^{i}(x), T^{i}(y)) : 0 ≤ i ≤ n − 1}.

For ϵ > 0, a set F ⊂ X is called (n, ϵ)-separated if it is separated with respect to the metric d_{n}, that is if d_{n}(x, y) ≥ ϵ for all distinct x, y ∈ F. With (F_{n}(ϵ))_{n∈ℕ} denoting a sequence of maximal (in the sense of inclusion) (n, ϵ)-separated sets, the topological pressure of the continuous potential function ϕ : X → IR is defined by

P(T, ϕ) := lim_{ϵ→0} lim sup_{n→∞} \frac{1}{n} log ( ∑_{x∈F_{n}(ϵ)} exp ∑_{j=0}^{n−1} ϕ ∘ T^{j}(x)) .

Note that the notion of topological pressure belongs to topological dynamics, whereas measure theoretical entropy h_{μ}(T) represents an important concept in ergodic theory. The link between these two notions is given by the following so called variational principle

P(T, ϕ) = sup{h_{μ}(T) + ∫ ϕ dμ},

where the supremum is taken with respect to all T -invariant (ergodic) Borel probability measures μ supported on X.

In the more general case of a GPL-map f, if there are critical points in the Julia set, then for t ≥ 0, the potential −t log |f′| is neither continuous nor bounded. Hence, a priori it is not clear how to adapt the above definition of pressure to this more general situation. However, for arbitrary rational maps F. Przytycki suggested in [6] several ways to extend the concept of topological pressure associated with the potential −t log |f′|. We now recall these suggestions in the setting of a GPL-map f, and for t ≥ 0.

(1) Variational pressure.

P_{V}(t) := sup{h_{μ}(f) − t ∫ log |f′|dμ},

where the supremum is taken with respect to all ergodic f-invariant measures supported on J(f).

(2) Hyperbolic variational pressure.

P_{HV}(t) := sup{h_{μ}(f) − t ∫ log |f′|dμ},

where the supremum is taken with respect to all ergodic f-invariant measures supported on J(f) such that the Lyapunov exponent is positive, i.e. such that ∫ log |f′|dμ > 0.

(3) Hyperbolic pressure.

P_{H}(t) := sup{P(f|_{X}, −t log |f′|)},

where the supremum is taken with respect to all f-invariant hyperbolic subsets X of J(f) such that some iterate of f|_{X} is topologically conjugate to a subshift of finite type.

(Recall that a forward invariant compact set X ⊂ J(f) is called hyperbolic if there exists n ≥ 1 such that |(f^{n})′(x)| > 1, for each x ∈ X).

(4) DU-pressure.

P_{DU}(t) := sup{P(f|_{K(V)}, −t log |f′|)},

where the supremum is taken with respect to all open subsets V of J(f) for which J(f) ∩ Crit(f) ⊂ V, and where K(V) := J(f) \ ⋃_{n≥0} f^{−n}(V). Note that K(V) is compact, f-invariant and disjoint from Crit(f).

(5) Conformal pressure.

P_{C}(t) := log λ(t),

where λ(t) is defined as the infimum of the set of all positive λ for which there exists a Borel probability measure m with the property d(m ∘ f)/dm = λ|f′|^{t}.

(6) Point pressure.

P_{z}(t) := lim sup_{n→∞} \frac{1}{n} log ∑_{x∈f^{−n}(z)} |(f^{n})′(x)|^{−t}

for z ∈ G, where G ⊂ \overline{C I} \ ⋃_{n≥0} f^{n}(Crit(f)) has the property that HD(\overline{C I} \ G) = 0 and that P_{z}(t) = P_{w}(t) for all z, w ∈ G. Note that the existence of such a set G has been obtained in [6].

Since, as we already mentioned, a parabolically semihyperbolic GPL-map is in particular non-recurrent, the following theorem can clearly be applied to the situation of a parabolically semihyperbolic GPL-map. For the proof of this theorem we refer to the proof of Theorem 2.6 in [8] (cf. also [6]).

Theorem 2.1. For a non-recurrent map f and for each t ≥ 0, all types of pressure functions defined in (1)-(6) above coincide. Their common value will be denoted by P(t). Furthermore, the following holds.

(a) If 0 ≤ t < HD(J(f)), then P(t) > 0.

(b) If Ω = ∅, then P(t) < 0 for all t > HD(J(f)).

(c) If Ω ≠ ∅, then P(t) = 0 for all t ≥ HD(J(f)).

Finally, we collect a few facts concerning conformal Gibbs states which will be relevant throughout. For the proofs of these facts we refer to [8] where they were in fact proven for non-recurrent rational functions. We remark that it is straight forward to adapt these proofs to the setting of a non-recurrent GPL-map. Recall that for a given t ≥ 0 a Borel probability measure m_{t} supported on J(f) is called t-conformal Gibbs state if f is non-singular with respect to m_{t} and

\frac{d(m_{t} ∘ f)}{dm_{t}} = e^{P(t)}|f′|^{t}.

In view of the definition of conformal pressure, it follows from Theorem 2.1 that for a nonrecurrent GPL-map and for each t ≥ 0 there exists a t-conformal Gibbs state m_{t}. Now, the following result is an immediate consequence of the definition of a t-conformal Gibbs state and the fact that m_{t} (⋃_{n=1}^{∞} f^{n}(Crit(f))) = 0.

Proposition 2.2. If f is a non-recurrent GPL-map f and P(t) > −χ_{0}t, then we have that m_{t} (\overline{⋃_{n≥1} f^{n}(Crit(f))}) = 0. (We remark that this conclusion can hence be drawn in particular for t ∈ (0, HD(J(f)))).

Furthermore, we have the following important result, which guarantees that even in the parabolic situation there exists a finite invariant measure equivalent to a conformal Gibbs state.

Theorem 2.3. ([8]) Let m_{t} be a t-conformal Gibbs state for a non-recurrent GPL-map f such that m_{t} (\overline{⋃_{n≥1} f^{n}(Crit(f))}) = 0. Then up to a multiplicative constant there exists a unique finvariant, σ-finite measure μ_{t} which is conservative, ergodic and equivalent to m_{t}. Moreover, if t ∈ [0, HD(J(f)))) then μ_{t} is a finite measure (which hence can always be assumed to be a probability measure).

3. Conformal graph directed Markov system and GPL-maps

Before we state the main result of this section, we first recall the definition of a conformal graph directed Markov system, abbreviated as CGDM-system. Recall from [5] that the combinatorical spine of a graph directed Markov system is represented by a directed multigraph (V, E, i, t, A), consisting of a finite set V of vertices, a countable set E of directed edges, two functions i, t : E → V, and a transition matrix A : E × E → {0, 1}. Here, i(e) refers to the initial vertex and t(e) to the terminal vertex of an edge e ∈ E. In our special context here, the matrix A = (A_{uv}) has the property that A_{uv} = 1 if and only if t(u) = i(v). We then define the corresponding symbolic space as follows.

E := {(e_{1}, e_{2}, …) ∈ E^{∞} : A_{e_{i}e_{i+1}} = 1 for all i ≥ 1}.

Additionally, assume that we are given a set {X_{v} : v ∈ V } of non-empty compact connected subsets X_{v} of C I, and a set Φ = {ϕ_{e} : X_{t(e)} → X_{i(e)}}_{e∈E} of univalent contractions, all with some fixed Lipschitz constant 0 < s < 1, which have respective conformal extensions from an open connected neighbourhood W_{t(e)} of X_{t(e)} to an open connected neighbourhood W_{i(e)} of X_{t(e)}. If Φ satisfies additionally the ‘open set condition’ and the ‘cone condition’ (see [5], section 4.2), then Φ is called a CGDM-system. The limit set J_{Φ} of Φ is then defined as follows. For an arbitrary τ = (τ_{1}, τ_{2}, …) ∈ E and n ≥ 1, let

ϕ_{τ|_{n}} := ϕ_{τ_{1}} ∘ … ∘ ϕ_{τ_{n}} : X_{t(τ_{n})} → X_{τ(τ_{1})}.

Since Φ consists of s-Lipschitz contractions, the intersection ⋂_{n≥1} ϕ_{τ|_{n}}(X_{t(τ_{n})}) is a singleton, which we denote by π(τ). In this way we obtain a map π : E → ⋃_{v∈V} X_{v}, and we let

J_{Φ} := π(E).

The following proposition will turn out to be crucial in our analysis of the analytic properties of the pressure function. Note that in the proof of this proposition we introduce some notation which will also be used in the following section. Also, in here ‘finitely primitive of order 2’ refers to that for each pair u, v ∈ V_{f} there exist a, b ∈ E_{f} such that i(a) = u, t(b) = v and A_{a,b} = 1 (cf. [5]).

Proposition 3.1. For a parabolically semihyperbolic GPL-map f there exists a CGDM-system Φ_{f} which is finitely primitive of order 2, such that J_{Φ_{f}} ⊂ J(f) and

J_{Φ_{f}} ∩ U_{o} = J(f) ∩ U_{o} \ ⋃_{n≥0} f^{−n}(Ω ∪ ⋂_{k≥0} f^{−k} (U_{r})) .

Proof. For the proof it is sufficient to show how to associate to f a CGDM-system. For this we define U_{(i,j)} := f_{j}^{−1}(U_{i}), for each (i, j) ∈ (I_{p} × I_{r}) ∪ (I_{p} × I_{p} \ {diag.}). Here {diag.} denotes the diagonal in I_{p} × I_{p}, and f_{j}^{−1} : U → U_{j} refers the inverse of the map f|_{U_{j}}. Using condition (c) in the definition of a parabolically semihyperbolic GPL-map, it follows that

U_{(i,j)} ∩ ⋃_{n≥1} f^{n}(Crit(f)) = ∅.  (3.1)

Let V_{f} := I_{o} ∪ (I_{p} × I_{r}) ∪ (I_{p} × I_{p} \ {diag.}) be the set of vertices. The conformal univalent contractions of our system are given as follows. By (3.1) and the definition of the set I_{o}, we have that for each v ∈ V_{f} the holomorphic inverse branches of any iterate of f are well-defined on U_{v}. Hence, for v ∈ V_{f} and n ≥ 1 we consider all holomorphic inverse branches f_{∗}^{−n} : U_{v} → U of f^{n} for which f_{∗}^{−n}(U_{v}) ⊂ U_{w} for some w ∈ V_{f}, and for which f^{k}(f_{∗}^{−n}(U_{v})) ∩ (⋃_{s∈V} U_{s}) = ∅ for all 1 ≤ k < n. In this situation we write ϕ_{e} : U_{t(e)} → U_{i(e)} instead of f_{∗}^{−n} : U_{v} → U_{w}, where t(e) = v and i(e) = w. Also, we define N(e) := n. Now, let

Φ_{f} := \{ϕ_{e} : \overline{U_{t(e)}} → \overline{U_{i(e)}}\}_{e∈E_{f}} ,

where E_{f} is some countable auxiliary set parametrizing the family Φ_{f}. Note that the set V_{f} of vertices is finite, whereas in general the set E_{f} of edges is infinite. We denote by E_{f} the corresponding symbolic space. Since \overline{U_{v}} ∩\overline{⋃_{n≥1} f^{n}(Crit(f))} = ∅, it follows that for each v ∈ V_{f} there exists an open connected simply connected set \overline{U_{v}} ⊂ W_{v} ⊂ U such that if e ∈ E_{f} and t(e) = v, then ϕ_{e} has a univalent holomorphic extension to W_{v} and ϕ_{e}(W_{v}) ⊂ U_{i(e)} (for later use, we also introduce accordingly W and W_{o} := ⋃_{i∈I_{o}} W_{i}). Since for each i ∈ I_{p} we have that ⋂_{n≥0} f^{−n}(J(f) ∩ \overline{U_{i}}) = \{ω_{i}\}, we immediately obtain from the construction of Φ_{f} that

J_{Φ_{f}} ∩ U_{o} = J(f) ∩ U_{o} \ ⋃_{n≥0} f^{−n}(Ω ∪ ⋂_{k≥0} f^{−k} (U_{r})) .

We remark that the cone condition is satisfied, since for each v ∈ V the boundaries of the disc \overline{U_{v}} is smooth. Also, the open set condition follows immediately from the construction of Φ_{f}, noting that the elements of Φ_{f} are inverse branches of forward iterates of f. Finally, since for each pair u, v ∈ V there exist a, b ∈ E_{f} such that i(b) ∈ I_{o} and such that i(a) = u, t(b) = v and A_{a,b} = 1, it follows that the system Φ_{f} is finitely primitive of order 2.

4. Real analyticity of the topological pressure

In this section we give the proof of our main result in this paper. From now on we shall always assume that f is a parabolically semihyperbolic GPL-map.

Lemma 4.1. For each positive t for which P(t) > −χ_{0}t, there exists 0 < ρ < 1 such that for all n ≥ 1 we have

m_{t} ( ⋂_{j=0}^{n} f^{−j}(U_{r})) ≪ ρ^{n}.

Proof. Fix q ≥ 1, and consider the set

U_{r}^{(q)} := U_{r} ∩ f^{−1}(U_{r}) ∩ … ∩ f^{−q}(U_{r}).

Since the map f : U_{j} → U is univalent for each j ∈ I_{r}, it follows by induction that there exist finitely many, say k_{q}, holomorphic inverse branches of f^{q}, denoted by f_{1}^{−q} : U → U_{r}, …, f_{k_{q}}^{−q} : U → U_{r}, such that

U_{r}^{(q)} = ⋃_{j=1}^{k_{q}} f_{j}^{−q}(U_{r}).  (4.1)

Hence, for any arbitrary set A ⊂ U_{r} it follows that

U_{r}^{(q)} ∩ f^{−q}(A) = ⋃_{j=1}^{k_{q}} f_{j}^{−q}(A),  (4.2)

and by conformality of m_{t} we have for each j ∈ {1, 2, …, k_{q}} that

m_{t}(f_{j}^{−q}(A)) ≤ m_{t}(A)e^{−P(t)q} sup_{z∈A}{|(f_{j}^{−q})′(z)|}^{t} ≤ m_{t}(A)e^{−P(t)q} sup_{z∈U_{r}}{|(f_{j}^{−q})′(z)|}^{t}, (4.3)

as well as

m_{t}(f_{j}^{−q}(U_{r})) ≥ m_{t}(U_{r})e^{−P(t)q} inf_{z∈U_{r}}{|(f_{j}^{−q})′(z)|}^{t}.  (4.4)

Now, on U_{r} we can apply Koebe’s distortion theorem, that is there exists a constant K ≥ 1 such that

sup_{z∈U_{r}}{|(f_{j}^{−q})′(z)|} ≤ K inf_{z∈U_{r}}{|(f_{j}^{−q})′(z)|}.

Therefore, (4.3) and (4.4) imply that

m_{t}(f_{j}^{−q}(A)) ≤ \frac{K^{t}}{m_{t}(U_{r})}m_{t}(A)m_{t}(f_{j}^{−q}(U_{r})).

Combining this estimate with (4.1) and (4.2), it follows that

m_{t}(U_{r}^{(q)} ∩ f^{−q}(A)) ≤ \frac{K^{t}}{m_{t}(U_{r})}m_{t}(U_{r}^{(q)})m_{t}(A).  (4.5)

Let U_{r}^{(∞)} := ⋂_{j≥0} f^{−j}(U_{r}) = ⋂_{q≥1} U_{r}^{(q)}, and observe that f^{−1}(U_{r}^{(∞)}) ⊃ U_{r}^{(∞)}. By ergodicity of μ_{t}, we hence have that μ_{t}(U_{r}^{(∞)}) ∈ {0, 1}. Now, since μ_{t}(U_{o}) > 0, and since U_{r} ⊂ U \ U_{o}, we have μ_{t}(U_{r}) < 1, which then implies that μ_{t}(U_{r}^{∞}) = 0. Since {U_{r}^{(q)}}_{q=1}^{∞} is a descending sequence of sets, we conclude that lim_{q→∞} μ_{t}(U_{r}^{(q)}) = 0, and hence that lim_{q→∞} m_{t}(U_{r}^{(q)}) = 0. Therefore, we can choose q ≥ 1 sufficiently large such that K^{t}m_{t}(U_{r}^{(q)})/m_{t}(U_{r}) ≤ 1/2. Inserting this observation into (4.5), we obtain that for any arbitrary A ⊂ U_{r} we have that

m_{t}(U_{r}^{(q)} ∩ f^{−q}(A)) ≤ \frac{1}{2} m_{t}(A).  (4.6)

In order to finish the proof, we use (4.6) and observe that for every k ≥ 1 we have that

m_{t} ( ⋂_{j=0}^{qk} f^{−j}(U_{r})) = m_{t} (U_{r}^{(q)} ∩ f^{−q} ( ⋂_{j=0}^{q(k−1)} f^{−j}(U_{r}))) ≤ \frac{1}{2}m_{t} ( ⋂_{j=0}^{q(k−1)} f^{−j}(U_{r})) .

By way of induction, this gives that

m_{t} ( ⋂_{j=0}^{qk} f^{−j}(U_{r})) ≤ (\frac{1}{2})^{k} ,

which also holds for k = 0. Now let n ≥ 1 be given, and write n = qk + r, for 0 ≤ r < q and k ≥ 0. It follows that

m_{t} ( ⋂_{j=0}^{n} f^{−j}(U_{r})) ≤ m_{t} ( ⋂_{j=0}^{qk} f^{−j}(U_{r})) ≤ (\frac{1}{2})^{k} ≤ (\frac{1}{2})^{\frac{n}{q} −1} = 2 ((\frac{1}{2})^{\frac{1}{q}} )^{n} .

As an immediate consequence we derive the following corollary, which shows that for certain values of t the sets J(f) and J_{Φ_{f}} coincide m_{t}-almost everywhere on U_{o}.

Corollary 4.2. For each positive t for which P(t) > −χ_{0}t, we have that m_{t}(J_{Φ_{f}} ∩ U_{o}) = m_{t}(U_{o}) > 0.

Proof. By Proposition 3.1 we have J_{Φ_{f}} ∩ U_{o} = J(f) ∩ U_{o} \ ⋃_{n≥0} f^{−n}(Ω ∪ ⋂_{k≥0} f^{−k} (U_{r})). Since P(t) > −χ_{0}t, Proposition 2.2 implies that m_{t} has no atoms. Finally, by Lemma 4.1 we have that m_{t}(⋂_{k≥0} f^{−k}(U_{r})) = 0. Combining these three observations, the statement of the corollary follows.

Lemma 4.3. For each c ∈ Crit(J(f)) we have χ(c) > 0.

Proof. For every q ≥ 1, let f_{1}^{−q}, …, f_{k_{q}}^{−q} be the holomorphic inverse branches which we already considered at the beginning of the proof of Lemma 4.1. By Vitali’s theorem, the family \{f_{i}^{−q} : q ≥ 1, 1 ≤ i ≤ k_{q}\} is normal, and since J(f) ⊂ \overline{U}, this implies that each point of accumulation of this family is a function equal to some constant. Hence, each point of accumulation of the family of derivatives of these functions is the constant function equal to zero. Thus, there exists q ≥ 1 such that |(f_{i}^{−q})′(z)| ≤ 1/2, for all 1 ≤ i ≤ k_{q}, z ∈ U_{r}. Also note that, since ⋃_{k≥1} f^{k}(Crit(f)) ⊂ U_{r}, we have for each pair n, l ≥ 1 that there exists j ∈ {1, 2, …, k_{l}} such that f_{j}^{−l}(f^{n+l}(c)) = f^{n}(c). Now, let l ≥ 1 be fixed such that l = sq +r, for 0 ≤ r < q and s ≥ 0. For each n ≥ 1 we then have by the chain rule

|(f^{l})′(f^{n}(c))| ≥ 2^{s}M_{q} ≥ 2^{\frac{l}{q} −1}M_{q} = (2^{1/q})^{l} M_{q}/2,

where we have set M_{q} := min_{0≤j<q}{inf{|(f^{j})′(z)| : z ∈ U_{r}}}, which is strictly positive due to

the fact that Crit(f) ∩ \overline{U_{r}} = ∅. Hence, it follows that inf_{j≥1}\{|(f^{l})′(f^{j}(c))|\} ≥ (2^{1/q})^{l} M_{q}/2 for every l ≥ 1, which implies that χ(c) ≥ (log 2)/q > 0.

Lemma 4.4. For each positive t for which P(t) > −χ_{q}t, there exists l ≥ 1 such that for each Borel set A ⊂ U we have

m_{t}(f^{−1}(A)) ≪ (m_{t}(A))^{1/l}.

Proof. Using the t-conformality of m_{t}, it follows that the assertion holds for all Borel sets A ⊂ U such that A ∩ ⋃_{c∈Crit(J(f))} B(f(c), δ) = ∅, for some fixed positive δ. Hence, from now on let a Borel set A ⊂ B(f(c), δ) be fixed, for some c ∈ Crit(J(f)), with m_{t}(A) > 0 and where δ < dist(U_{r}, ∂U)/2 is chosen sufficiently small (which will be specified during the proof). Let f_{c}^{−1}(A) be the intersection of f^{−1}(A) with the component of f^{−1}(B(f(c), δ)) which contains c. Also, for n ≥ 1 we define

λ_{n}(c) := |(f^{n})′(f(c))|,

and let A(w, r, R) := {z ∈ C I : r ≤ |z − w| < R} denote the annulus centred at w ∈ C I of inner radius r and outer radius R. The structure of the proof is as follows. We show that for each pair s, k ≥ 1 we have that (i) m_{t}(B(c, (δλ_{sk}(c)^{−1})^{1/q(c)})) ≪ λ_{sk}(c)^{−t/q(c)}e^{−P(t)sk}. For this we slice B(f(c), δ) into annuli and define the ‘stopping time’

u := sup{n ≥ 0 : m_{t}(A ∩ A(f(c), δλ_{sn}(c)^{−1}, δ)) ≤ λ_{sn}(c)^{−t}e^{−P(t)sn}}.

We show that u is a finite number, and by combining this with the estimate in (i), we obtain (ii) m_{t}(f_{c}^{−1}(A)) ≪ λ_{su}(c)^{−t/q(c)}e^{−P(t)su}. Finally, we prove the following two facts, which then finishes the proof of the proposition. (iii) λ_{su}(c)^{−t/q(c)}e^{−P(t)su} ≤ (λ_{su}(c)^{−t}e^{−P(t)su})^{1/l}, for some l > 0 and for s sufficiently large.

(iv) λ_{su}(c)^{−t}e^{−P(t)su} ≪ m_{t}(A ∩ A(f(c), δλ_{s(u+1)}(c)^{−1}, δ)) ( ≤ m_{t}(A) ).

For (i), first note that by Koebe’s distortion theorem we have for all n ≥ 1 that

m_{t}(B(f(c), δλ_{n}(c)^{−1})) ≍ λ_{n}(c)^{−t}e^{−P(t)n}.

Using this observation and the fact that |(f_{c}^{−1})′(z)| ≍ |z − f(c)|^{−(1−1/q(c))} for z ≠ f(c) close to f(c) (where f_{c}^{−1} refers to an inverse branch of f defined on some neighbourhood of z which maps z close to c), it follows for each pair s, k ≥ 1 that

m_{t}(B(c, (δλ_{sk}(c)^{−1})^{1/q(c)})) = ∑_{j=k}^{∞} m_{t}(A(c, (δλ_{s(j+1)}(c)^{−1})^{1/q(c)}, (δλ_{sj}(c)^{−1})^{1/q(c)}))

≍ ∑_{j=k}^{∞} m_{t}(f_{c}^{−1}(A(f(c), δλ_{s(j+1)}(c)^{−1}, δλ_{sj}(c)^{−1})))

≍ ∑_{j=k}^{∞} λ_{sj}(c)^{(1− \frac{1}{q(c)})t}e^{−P(t)}m_{t}(A(f(c), δλ_{s(j+1)}(c)^{−1}, δλ_{sj}(c)^{−1}))

≤ e^{−P(t)} ∑_{j=k}^{∞} λ_{sj}(c)^{(1− \frac{1}{q(c)})t}m_{t}(B(f(c), δλ_{sj}(c)^{−1}))

≍ ∑_{j=k}^{∞} λ_{sj}(c)^{(1− \frac{1}{q(c)})t}λ_{sj}(c)^{−t}e^{−P(t)sj} = ∑_{j=k}^{∞} λ_{sj}(c)^{−t/q(c)}e^{−P(t)sj}

= λ_{sk}(c)^{−t/q(c)}e^{−P(t)sk} (1 + ∑_{j=k+1}^{∞} (\frac{λ_{sj}(c)}{λ_{sk}(c)})^{−t/q(c)} e^{−P(t)s(j−k)}) .

Hence, we are left with to show that the sum in the latter expression is bounded from above. For this recall that, since χ(c) > −q(c)P(t)/t, there exists κ > 0 and some s depending on c, such that \frac{1}{v} log |(f^{v})′(f^{n}(c))| ≥ −q(c)P(t)/t + κ for all for all v ≥ s, n ≥ 1. Thus, for every j ≥ k + 1 we have that

log (\frac{λ_{sj}(c)}{λ_{sk}(c)}) = ∑_{i=k}^{j−1} log λ_{s(i+1)}(c) − log λ_{si}(c) = ∑_{i=k}^{j−1} log (\frac{λ_{s(i+1)}(c)}{λ_{si}(c)})

= ∑_{i=k}^{j−1} log |(f^{s})′(f^{si}(c))| ≥ (\frac{−q(c)P(t)}{t} + κ) s(j − k).

It follows that

(\frac{λ_{sj}(c)}{λ_{sk}(c)})^{−t/q(c)} e^{−P(t)s(j−k)} ≤ e^{−(−q(c)P(t)+tκ)s(j−k)/q(c)}e^{−P(t)s(j−k)} = exp (−\frac{tκs}{q(c)}(j − k)) ,

which completes the proof of the statement in (i) above.

Since lim_{n→∞} m_{t}(A ∩ A(f(c), δλ_{sn}(c)^{−1}, δ) = m_{t}(A) > 0, in order to see that the ‘stopping time’ u is finite it is sufficient to show that

lim_{n→∞} λ_{sn}(c)^{−t}e^{−P(t)sn} = 0.

If P(t) ≥ 0, then this is an immediate consequence of Lemma 4.3. If P(t) < 0, then tχ(c) > −q(c)P(t) implies that t log |(f^{v})′(f(c))| > −vq(c)P(t), for each v sufficiently large. This gives that λ_{v}(c)^{−t}e^{−P(t)v} < e^{P(t)v(q(c)−1)}, and since q(c) ≥ 2, the result follows.

For (ii), we combine (i) and the finiteness of u and obtain

m_{t}(f_{c}^{−1}(A)) = m_{t}(f_{c}^{−1}(A ∩ \overline{B(f(c), δλ_{su}(c)^{−1})})) + m_{t}(A ∩ A(f(c), δλ_{su}(c)^{−1}), δ))
≤ m_{t}(f_{c}^{−1}(\overline{B(f(c), δλ_{su}(c)^{−1})})) + m_{t}(A ∩ A(f(c), δλ_{su}(c)^{−1}), δ))

≪ m_{t} (\overline{B (c, (δλ_{su}(c)^{−1})^{1/q(c)})}) + (δλ_{su}(c)^{−1})^{(\frac{1}{q(c)} −1)t}m_{t}(A ∩ A(f(c), δλ_{su}(c)^{−1}, δ))

≤ K_{c}λ_{su}(c)^{−t/q(c)}e^{−P(t)su} + (δλ_{su}(c)^{−1})^{(\frac{1}{q(c)} −1)t}e^{−P(t)su}λ_{su}(c)^{−t}
≍ λ_{su}(c)^{−t/q(c)}e^{−P(t)su},

where we have set K_{c} := (1 + ∑_{j=k+1}^{∞} exp(−tκs(j − k)/q(c))).

For (iii), recall that \frac{1}{v} log |(f^{v})′(f(c))| ≥ −q(c)P(t)/t + κ for all for all v ≥ s. Hence, by choosing l(c) sufficiently large such that κ > P(t)q(c)(1 − q(c))/(t(l(c) − q(c)), it follows that

(l(c) − q(c))\frac{log λ_{su}(c)}{su} ≥ (\frac{−q(c)P(t)}{t} + κ) (l(c) − q(c)) ≥ \frac{q(c)P(t)}{t}(1 − l(c)).

An elementary rearrangement then gives

λ_{su}(c)^{−t/q(c)}e^{−P(t)su} ≤ (λ_{su}(c)^{−t}e^{−P(t)su})^{1/l(c)}.

By defining l := max{l(c) : c ∈ Crit(J(f))}, the statement in (iii) follows.

Finally for (iv), the finiteness of u gives

m_{t}(A ∩ A(f(c), δλ_{s(u+1)}(c)^{−1}, δ) > λ_{s(u+1)}(c)^{−t}e^{−P(t)s(u+1)} ≥ e^{−P(t)s}||f′||^{−st}λ_{su}(c)^{−t}e^{−P(t)su},

which completes the proof of the proposition.

We now pass to the CGDM-system Φ_{f} associated with the GPL-map f. For this the reader is asked to recall the construction and notation given in Section 3. For each t ≥ 0, s ∈ IR and e ∈ E_{f} we define the potential g_{t,s}^{(e)} : W_{t(e)} → IR for x ∈ W_{t(e)} by

g_{t,s}^{(e)}(x) := t log |ϕ′_{e}(x)| − sN(e).

We shall now see that for suitably chosen s and t the family G_{t,s} := {g_{t,s}^{(e)} : e ∈ E_{f}} is a summable Hölder family of functions, where Hölder refers to the fact that for some γ > 0 we have (cf. [2], [5])

sup_{n≥1} sup_{(τ_{1},τ_{2},...)∈E_{f}} sup_{z,w∈U_{t(τn)}} |g_{t,s}^{(τ_{1})}(ϕ_{τ_{2},... ,τ_{n}}(z)) − g_{t,s}^{(τ_{1})}(ϕ_{τ_{2},... ,τ_{n}}(w))|e^{γ(n−1)} < ∞.

Lemma 4.5. For each positive u such that P(u) > −χu, there exists δ > 0 such that G_{t,s} is a summable Hölder family of functions, for each t ∈ (u − δ, u + δ) and s > P(u) − δ.

Proof. Using Koebe’s distortion theorem, it is straight forward to see that G_{t,s} is a Hölder family of functions, for each t ≥ 0 and s ∈ IR (see [5] paragraph 4.2, Lemma 2.2). In order to prove that G_{t,s} is summable, put Z^{(n)} := {e ∈ E_{f} : N(e) = n} and define

R_{n} := ⋃_{e∈Z^{(n)}} ϕ_{e}(U_{t(e)}) for n ≥ 1.

We first observe that if there are no parabolic elements then we have for n > 1 that R_{n} ⊂ f^{−1} (⋂_{j=0}^{n−2} f^{−j}(U_{r})) (for n = 1, we have R_{1} ⊂ U_{o}), and hence Lemma 4.1 and Lemma 4.4 imply

m_{u}(R_{n}) ≤ m_{u} (f^{−1} ( ⋂_{j=0}^{n−2} f^{−j}(U_{r}))) ≪ (m_{u} ( ⋂_{j=0}^{n−2} f^{−j}(U_{r})))^{1/l} ≪ ρ^{n/l}.  (4.7)

If there are parabolic points then χ_{0} = 0, and consequently the condition P(u) > −χu implies that P(u) > 0. For e ∈ Z^{(n)} we have that there exists 1 ≤ k ≤ n such that f^{j}(U_{i(e)}) ⊂ U_{r} for all k ≤ j < n and such that f^{j}(U_{i(e)}) ⊂ U_{i} for all 1 ≤ j < k and for some i ∈ I_{p}. Using Lemma 4.1, Lemma 4.4 and (LBP), we obtain in this situation, with some fixed β such that max{e^{−P(u)}, ρ} < β < 1,

\begin{aligned}m_{u}(R_{n}) & ≪ (m_{u}(f(R_{n})))^{1/l} ≪ (∑_{k=1}^{n} e^{−kP(u)} ∑_{i∈I_{p}} k^{− \frac{p_{i}+1}{p_{i}} u}m_{u} ( ⋂_{j=1}^{n−k} f^{−j}(U_{r})))^{1/l} \\ & ≪ (e^{−kP(u)}ρ^{n−k} card(I_{p}))^{1/l} ≪ β^{n/l}. \end{aligned}

Combining this estimate and (4.7), we conclude that no matter if there are parabolic points or not, there exists α > 0 such that for all n ≥ 1 we have

m_{u}(R_{n}) ≪ e^{−αn}.  (4.8)

Using the definition of the measure m_{u} along with Koebe’s distortion theorem, we now immediately have for all n ≥ 1 that

∑_{e∈Z^{(n)}} sup_{z∈\overline{U_{i(e)}}}(|(f^{n})′(z)|^{−u})e^{−P(u)n} ≪ e^{−αn}.

Observe that the family {ϕ_{e} : W_{t(e)} → U_{i(e)}}_{e∈E_{f}} is normal, and hence all its limit functions are constant. This implies that

Δ_{1} := sup_{e∈E_{f}} sup_{z∈\overline{U_{t(e)}}} |ϕ′_{e}(z)| < ∞.

Therefore, for fixed u > 0 there exists 0 < δ < min\{u, \frac{α}{2}, \frac{α}{4}| log Δ_{1}|^{−1}, \frac{α}{4}| log Δ_{2}|^{−1}\}, where we have put Δ_{2} := sup_{e∈E_{f}} sup_{z∈\overline{U_{i(e)}}} |f′(z)|. With this choice of δ we obtain for each t ∈ (u − δ, u + δ) and s > P(u) − δ that

∑_{e∈Z^{(n)}} sup_{z∈\overline{U_{i(e)}}}(|(f^{n})′(z)|^{−t})e^{−sn} ≤ ∑_{e∈Z^{(n)}} sup_{z∈\overline{U_{i(e)}}}(|(f^{n})′(z)|^{−u}) e^{−P(u)n}e^{δn} max\{Δ_{1}^{δ}, Δ_{2}^{nδ}\}
≪ e^{−αn}e^{\frac{α}{2} n}e^{\frac{α}{4} n} = e^{− \frac{α}{4} n}.

For the following lemma recall that the topological pressure P associated with the family G_{t,s} is given by (cf. [2], [5])

P(t, s) := lim_{n→∞} \frac{1}{n} log ∑_{(τ_{1},... ,τ_{n})∈E_{f}^{(n)}} sup_{z∈U_{t(τn)}} exp (g_{t,s}^{(τ_{n})}(z) + ∑_{i=1}^{n−1} g_{t,s}^{(τ_{i})}(ϕ_{τ_{i+1},... ,τ_{n}}(z))) ,

where we have set E_{f}^{(n)} := {(τ_{1}, …, τ_{n}) ∈ E_{f}^{n} : A_{τ_{j}τ_{j+1}} = 1 for all j = 1, 2, …, n − 1}. Also, associated with G_{t,s} there exists a unique G_{t,s}-conformal probability measure m_{t,s} supported on J_{Φ_{f}}. That is, for each n ≥ 1 and τ = (τ_{1}, …, τ_{n}) ∈ E_{f}^{(n)} we have for every Borel set A ⊂ U_{t(τ_{n})} that

m_{t,s}(ϕ_{τ}(A)) = ∫_{A} exp (g_{t,s}^{(τ_{n})}(z) + ∑_{i=1}^{n−1} g_{t,s}^{(τ_{i})}(ϕ_{τ_{i+1},... ,τ_{n}}(z)) − nP(t, s)) dm_{t,s}(z).

Lemma 4.6. For each positive t for which P(t) > −χt, we have P(t, P(t)) = 0. Furthermore, for every n ≥ 1 and for each τ = (τ_{1}, …, τ_{n}) ∈ E_{f}^{(n)} we have that

m_{t,P(t)}(ϕ_{τ}(U_{t(τ_{n})})) ≍ m_{t}(ϕ_{τ}(U_{t(τ_{n})})),

with comparability constants not depending on n and τ.

Proof. By conformality of m_{t} and m_{t,s}, we have for each n ≥ 1 that

m_{t}(ϕ_{τ}(U_{t(τ_{n})})) = ∫_{U_{t(τn)}} |ϕ′_{τ}(z)|^{t}e^{−P(t) ∑_{j=1}^{n} N(τ_{j})}dm_{t}(z)
≍ ||ϕ′_{τ}||^{t}e^{−P(t) ∑_{j=1}^{n} N(τ_{j})}m_{t}(U_{t(τ_{n})})
≍ e^{nP(t,P(t))}||ϕ′_{τ}||^{t}e^{−P(t) ∑_{j=1}^{n} N(τ_{j})}e^{−nP(t,P(t))}
≍ e^{nP(t,P(t))}m_{t,P(t)}(ϕ_{τ}(U_{t(τ_{n})})).

Therefore, if on the one hand P(t, P(t)) > 0 then m_{t,P(t)}(J_{Φ_{f}}) = 0, which contradicts m_{t,P(t)}(J_{Φ_{f}}) = 1. On the other hand, if P(t, P(t)) < 0 then we obtain m_{t}(J_{Φ_{f}}) = 0, which is also a contradiction. Thus, it follows that P(t, P(t)) = 0, which gives the lemma.

Proof of the Theorem 1.1. Using Lemma 4.5 and applying Theorem 2.6.12 of [5] (or alternatively [2] Theorem 6.4), we have for each positive u with P(u) > −χu that there exists δ > 0 such that P is real-analytic on (u − δ, u + δ) × (P(u) − δ, P(u) + δ) in both variables t and s. In order to prove that P is real-analytic on (u − δ, u + δ), we employ the implicit function theorem, showing that P is the unique real-analytic function which satisfies P(t, P(t)) = 0 for all t ∈ (u − δ, u + δ). For this it is now sufficient to verify that for all t ∈ (u − δ, u + δ) we have

\frac{∂P(t, s)}{∂s}|_{(t,P(t))} exists and is strictly negative.  (4.9)

Denote the measure m_{t,P(t)} by ν_{t}. Proposition 3.1, Lemma 4.5 and Lemma 4.6 guarantee that Theorem 3.7 of [5] is applicable. This gives that the measure ν_{t} has a lift ν̃_{t} to the symbolic space E_{f}, and that there exists a measure μ̃_{t} in the measure class of ν̃_{t} which is invariant under the shift map on the space E_{f}, and whose Radon-Nikodym derivative with respect to ν̃_{t} is bounded away from zero and infinity. We can now apply Proposition 2.6.13 of [5] (or alternatively [2] Proposition 6.5), which gives

\frac{∂P(t, s)}{∂s}|_{(t,P(t))} = − ∫ Ndμ̃_{t}.  (4.10)

Using the estimate in (4.8) and the second part of Lemma 4.6 we then compute

∫ Ndμ̃_{t} ≍ ∫ Ndν̃_{t} = ∫ Ndν_{t} = ∑_{n≥1} nν_{t}(R_{n}) ≍ ∑_{n≥1} nm_{t}(R_{n}) ≪ ∑_{n≥1} ne^{−αn} < ∞, (4.11)

where after the first equality sign we treated the function N slightly informally as defined on the limit set J_{Φ_{f}}. Combining (4.10) and (4.11), and using the fact that the function N is strictly positive, we derive (4.9), which then completes the proof of Theorem 1.1.

References

[1] J. Aaronson, M. Denker, M. Urbanski, ‘Ergodic theory for Markov fibred systems and parabolic rational maps’, Trans. of the AMS, Vol. 337, No. 2 (1993) page 495 - 548.

[2] P. Hanus, D. Mauldin, M. Urbański, ‘Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems’, preprint IHES 1999, to appear in Acta Math. Hungarica.

[3] N. Makarov, S. Smirnov, ‘On thermodynamics of rational maps II. Non-recurrent maps’, preprint 2000.

[4] D. Mauldin, M. Urbański, ‘Fractal measures for parabolic IFS’, reprint 2000, to appear in Adv. Math..

[5] D. Mauldin, M. Urbański, Graph directed Markov systems, lecture notes, preprint 2000.

[6] F. Przytycki, ‘Conical limit set and Poincaré exponent for iterations of rational functions’, Trans. A.M.S. 351 (1999), 2081-2099.

[7] S. Smirnov, ‘Spectral analysis of Julia sets’, thesis (1996).

[8] M. Urbański, ‘Thermodynamic formalism, topological pressure and escape rates for non-recurrent conformal dynamics’, preprint 2000.

[9] M. Urbański, M. Zinsmeister, ‘Geometry of hyperbolic Julia-Lavaurs sets’, preprint 2000, to appear in Indagationes Math..

Bernd O. Stratmann; Mathematical Institute, University of St Andrews, St Andrews

KY16 9SS, Scotland. bos@maths.st-and.ac.uk, http://www.maths.st-and.ac.uk/∼bos

Mariusz Urbański; Department of Mathematics, University of North Texas, P.O. Box

311430, Denton, TX 76203-1430, USA. urbanski@unt.edu, http://www.math.unt.edu/∼urbanski