Open AccessArticle

Bowen’s Formula for a Dynamical Solenoid

Andrzej Biś

Wojciech Kozłowski

and

Agnieszka Marczuk

Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Łódź, Poland

Author to whom correspondence should be addressed.

Entropy 2024, 26(11), 979; https://doi.org/10.3390/e26110979

Submission received: 19 July 2024 / Revised: 14 September 2024 / Accepted: 20 September 2024 / Published: 15 November 2024

(This article belongs to the Section Statistical Physics)

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Abstract

More than 50 years ago, Rufus Bowen noticed a natural relation between the ergodic theory and the dimension theory of dynamical systems. He proved a formula, known today as the Bowen’s formula, that relates the Hausdorff dimension of a conformal repeller to the zero of a pressure function defined by a single conformal map. In this paper, we extend the result of Bowen to a sequence of conformal maps. We present a dynamical solenoid, i.e., a generalized dynamical system obtained by backward compositions of a sequence of continuous surjections

{(f_{n} : X \to X)}_{n \in N}

defined on a compact metric space

(X, d) .

Under mild assumptions, we provide a self-contained proof that Bowen’s formula holds for dynamical conformal solenoids. As a corollary, we obtain that the Bowen’s formula holds for a conformal surjection

f : X \to X

of a compact

Keywords:

topological pressure; conformal maps; topological entropy

1. Introduction and Statement of Results

In 1979, Rufus Bowen [1] noticed a natural connection between the geometric properties of a conformal repeller and the thermodynamic formalism of equation describing this repeller. Bowen proved that for a certain transformation f of the Riemann sphere into itself, there exists a compact f-invariant subset J called a quasi-circle. The Hausdorff dimension t of this quasi-circle is a unique root of the equation

P_{J} (- t \cdot φ) = 0,

where

P_{J}

is the topological pressure of the map

f : J \to J

, and

φ = log (| f^{'} (z) |)

is the geometric potential. In the mathematical literature, the equation

P_{J} (- t \cdot φ) = 0

is known as the Bowen’s formula or Bowen’s equation. In 1982, Ruelle [2] showed that the Hausdorff dimension of the Julia set of a uniformly hyperbolic rational map depends real-analytically on parameters. The Bowen’s equation has various generalizations and applications in both real and complex dynamics. For example, the result of Barański [3] for some hyperbolic meromorphic maps was generalized by Kotus and Urbański [4] to the case of so-called regular Walters expanding conformal maps. The thermodynamical formalism theory for hyperbolic maps in the exponential family was developed by Urbański and Zdunik [5,6]. Also there is a recent paper by Barański, Karpińska, and Zdunik [7] on Bowen’s formula for meromorphic functions. There are also known generalizations of Bowen’s formula to the case of semigroup; Jaerisch and Sumi [8] obtained Bowen’s formula for a pre-Julia set of a nicely expanding rational semigroups. In [9,10], one can find applications of Bowen’s equation in examples for expanding Markov maps. In 2008, Rugh [11] generalized Bowen’s result and proved the following result.

Theorem 1

(Rugh [11]). Let M be a Riemannian manifold,

V \subset M

be an open set,

f : V \to M

be a conformal map of class

C^{1 + ϵ},

and

J \subset V

be its repeller. Then, the equation

P_{J} (- t \cdot φ) = 0,

where

φ (x) = log (| | D f (x) | |),

has a unique root

t^{*}

equal to the Hausdorff dimension of the set

J .

In 2011, Climenhaga [12] presented a generalization of the Bowen’s formula to the case of a continuous map

f : X \to X

defined on a compact metric space

(X, d)

with a nonzero dilatation, denoted here by

a_{w} (x) .

In Climenhaga’s work, we encounter the following definition of conformality. A map

f : X \to X

is conformal with dilatation

a_{w} (x)

if for any

x \in X

there exists a limit

a_{w} (x) : = lim_{y \to x} \frac{d (f (x), f (y))}{d (x, y)}

and the map

a_{w} : X \to [0, \infty)

is continuous. However, this notion is not suitable for our approach. Therefore, in Section 2, we introduce and apply a notion of strong dilatation and strong conformality.

For a conformal map (respectively, for strong conformal map), we define the Birkhoff sum, lower and upper Lyapunov exponents as follows.

\begin{matrix} S_{n} (log a_{w}) (x) & = \sum_{k = 0}^{n - 1} log a_{w} (f^{k} (x)), \\ λ_{n} (x) & = \frac{1}{n} \cdot S_{n} (log a_{w}) (x) . \end{matrix}

The limits

\underset{̲}{λ} (x) = \underset{n \to \infty}{lim inf} λ_{n} (x)

are called the lower Lyapunov exponents, and

\bar{λ} (x) = \underset{n \to \infty}{lim sup} λ_{n} (x)

are the upper Lyapunov exponents. For a subset

E \subset R,

we denote by

A (E) = {x \in X : [\underset{̲}{λ} (x), \bar{λ} (x)] \subset E} .

Climenhaga introduced the set

B

as the set of points

x \in X

for which the condition

inf {S_{n - k} (log a_{w}) (f^{k} (x)) + n \cdot ε : 0 \leq k \leq n, n \in N} > - \infty

holds, and he proved the following theorem.

Theorem 2

(Climenhaga [12]). Let X be a compact metric space, and

f : X \to X

be a continuous conformal map with dilatation

a_{w} (x) .

Assume that f has no points of zero or infinite dilatation, meaning that for any

x \in X,

the inequalities

0 < a_{w} (x) < \infty

hold. Then, for any subset

Z \subset A (0, \infty) \cap B,

the unique root

t^{*}

of the pressure function

t \mapsto P_{Z} (- t \cdot log a_{w})

is equal to the Hausdorff dimension

d i m_{H} (Z)

Z .

The rest of this section is devoted to the Bowen’s formula for a dynamical solenoid, i.e., the sequence

f_{\infty} = {(f_{n} : X \to X)}_{n \in N}

of continuous surjections defined on a compact metric space

(X, d) .

In general, a sequence

f_{\infty} = {(f_{n} : X \to X)}_{n \in N}

of continuous surjections determines two distinct dynamical systems: nonautonomous dynamical system (when we consider forward compositions of

f_{n} : X \to X

) and dynamical solenoid (when we consider dynamics determined by inverse limit of the sequence). There are many papers on the dynamics of nonautonomous dynamical systems. The dynamics of dynamical solenoids, which are equally interesting, were studied less intensively (see, for example, [13,14]).

Additionally, we assume that

f_{\infty}

is expanding and strongly conformal; it means that each

f_{n} : X \to X

is a strongly conformal and expanding map (for precise definitions, see Section 2).

We also refer the reader to Section 3 where they will find detailed definitions of the sequence

a_{\infty}

, Lyapunov exponents

\underset{̲}{λ^{f_{\infty}}}, \bar{λ^{f_{\infty}}}

and

A (E)

in our case.

Our proof of the Bowen’s formula for a dynamical solenoid is inspired by Climenhaga’s paper, where he considered a continuous conformal map on a compact metric space. The main results of the paper are the following two theorems. In Theorem 3, we consider a topological entropy

h_{t o p} (f_{\infty}, Z) < \infty

of a dynamical solenoid

f_{\infty},

restricted to a subset

Z \subset X,

and provide estimation of a unique root of its pressure function

t \to P_{Z} (- t log a_{\infty})

Theorem 3.

Assume that

h_{t o p} (f_{\infty}, Z) < \infty

. Let

c, C \in R

with

c \leq C

, be such that for any

x \in Z

c \leq \underset{̲}{λ^{f_{\infty}}} (x) \leq \bar{λ^{f_{\infty}}} (x) \leq C .

c > 0

C < 0

, then there exists a unique

t^{'} \in [\frac{1}{C} h_{t o p} (f_{\infty}, Z), \frac{1}{c} h_{t o p} (f_{\infty}, Z)]

, such that

P (- t^{'} log a_{\infty}) = 0

. In particular, when

c = C \neq 0

, i.e., the Lyapunov exponent

λ^{f_{\infty}} \neq 0

is constant on Z, then

t^{'} = \frac{h_{t o p} (f_{\infty}, Z)}{λ^{f_{\infty}}} .

In this case,

P_{Z} (- \frac{h_{t o p} (f_{\infty}, Z)}{λ^{f_{\infty}}} log a_{\infty}) = 0 .

In case of a dynamical conformal simple and expanding solenoid, called for simplicity a DCSE-solenoid (see Section 5), we show that the Bowen’s formula holds, i.e., we prove the following result.

Theorem 4.

Let

(X_{\infty}, f_{\infty}, a_{\infty})

be a DCSE-solenoid defined on a compact metric space

(X, d)

, and let

Z \subset X

. Assume that

h_{t o p} (f_{\infty}, Z) < \infty

and

Z \subset A ((α, \infty))

for some

α > 0

. Then, there exists a unique root

t^{*}

of the pressure function

t \mapsto P_{Z} (- t \cdot log a_{\infty}))

with

{dim}_{H} (Z) = t^{*}

Corollary 1.

Let X be a compact metric space and

f : X \to X

be a strong conformal surjection with strong dilatation a. If f is regular and locally expanding, i.e.,

1 < a (x) < \infty

for any

x \in X

, then, for any subset

Z \subset A (α, \infty),

where

0 < α < \infty,

the unique root

t^{*}

of the pressure function

t \mapsto P_{Z} (- t \cdot log (a))

is equal to the Hausdorff dimension

d i m_{H} (Z)

Z .

The paper is organized as follows. In the Introduction, we provide the motivation of our research, the history of Bowen’s formula, and present main results of the paper. Section 2 is devoted to strong conformal maps and strong dilatation and basic properties of these notions. In Section 3, we introduce a notion of Lyapunov exponent, topological entropy, and topological pressure (in the spirit of Carathéodory structures elaborated by Pesin [15]) of a conformal dynamical solenoid. We prove basic properties of a pressure function of a strongly conformal dynamical solenoid. We also present a proof of Theorem 3. In Section 4, we describe the equivalence of two different approaches to the Hausdorff dimension. Section 5 is devoted to the proof of Lemma 11, the essential lemma used in the proof of Theorem 4. In Section 6, we prove Theorem 4. Finally, in the Section 7, we present an example showing that the notions of conformal map and of strongly conformal map are not equivalent.

2. Conformal Mappings

Let

(X, d)

be a metric space. For any

x \in X

and

δ > 0

, let

B (x, δ)

denote the open ball centered at point x with radius

δ

. We will say that the mapping

f : X \to X

is strong conformal if there exists a continuous function

a : X \to [0, \infty]

, called a strong dilatation, such that for any

x_{0} \in X

lim_{\begin{matrix} x \to x_{0} \\ y \to x_{0} \\ x \neq y \end{matrix}} \frac{d (f (y), f (x))}{d (y, x)} = a (x_{0}) .

(1)

If, for any

x \in X

0 < a (x) < \infty

, we call f a strong conformal regular mapping.

Lemma 1.

f : X \to X

is a strong conformal mapping with strong dilatation

a < \infty

, then f is locally Lipschitz, i.e., for any

x_{0} \in X

, there exists

δ > 0

and a constant L such that

d (f (x), f (y)) \leq L d (x, y), x, y \in B (x_{0}, δ) .

In particular, f is continuous.

Proof.

Let

x_{0} \in X

. Since

a (x_{0}) < \infty

and (1), there exists

δ > 0

such that for

x, y \in B (x_{0}, δ)

, the inequality

d (f (y), f (x)) < L \cdot d (y, x)

is satisfied with

L = a (x_{0}) + 1

. This completes the proof. □

Corollary 2.

f : X \to X

is a strong conformal mapping with strong dilatation

0 < a < \infty

and X is a compact metric space, then f is Lipschitz on X. In particular, f is uniformly continuous.

Proof.

This follows directly by Lemma 1 and the fact that any locally Lipschitz function is Lipschitz on any compact set. □

Lemma 2.

Assume that

f : X \to X

is a strong conformal mapping with strong dilatation a. Let

x_{0} \in X

. If

a (x_{0}) > 0

, then there exists

ε > 0

such that

f (y) \neq f (x)

whenever

x, y \in B (x_{0}, ε)

and

y \neq x

. In other words, f is a one-to-one mapping in some neighborhood of the point

x_{0}

Proof.

Let

A \in (0, a (x_{0}))

. Such A exists due to

a (x) > 0

. From the condition (1), it follows that there exists

ε > 0

such that

d (f (y), f (x)) > A \cdot d (y, x) > 0

for any

x, y \in B (x_{0}, ε)

and

y \neq x

. □

Corollary 3.

Every strong regular conformal mapping is continuous and locally one-to-one.

Proposition 1.

Let

f : X \to X

be a strong regular conformal mapping defined on a compact metric space

(X, d)

, and let a denote the strong dilatation of f. Then f is uniformly locally one-to-one, i.e., there exists

\hat{δ} > 0

such that for any

U \subset X

with diameter

< \hat{δ}

f | U

is one-to-one.

Proof.

By Corollary 3 there exists a finite cover

U = {U_{1}, \dots, U_{n}}

of X, such that for any

i = 1, \dots, n

f | U_{i}

is one-to-one. Let

\hat{δ}

be the Lebesgue number of the cover

U

. Then, for any set U with diameter

< \hat{δ}

U \subset U_{i}

for some i. Consequently,

f | U

is one-to-one. □

Lemma 3.

Assume that

f_{i} : X \to X

i = 1, 2

, are strong regular conformal mappings with strong dilatations

a_{i}

. Then, the composition

f_{1} \circ f_{2} : X \to X

is a continuous strong regular conformal mapping with strong dilalation

a = (a_{1} \circ f_{2}) \cdot a_{2}

, i.e.,

a (x_{0}) = a_{1} (f_{2} (x_{0})) \cdot a_{2} (x_{0}) .

Proof.

Notice that

f_{1} \circ f_{2} : X \to X

is continuous as the composition of continuous mappings. Let

x_{0} \in X

be fixed. By Lemma 2,

f_{2}

is one-to-one in some neighborhood of

x_{0}

. Thus, we obtain

\begin{matrix} lim_{\begin{matrix} x \to x_{0} \\ y \to x_{0} \\ x \neq y \end{matrix}} \frac{d (f_{1} (f_{2} (y)), f_{1} (f_{2} (x)))}{d (y, x)} \\ = & lim_{\begin{matrix} x \to x_{0} \\ y \to x_{0} \\ x \neq y \end{matrix}} \frac{d (f_{1} (f_{2} (y)), f_{1} (f_{2} (x)))}{d (f_{2} (y), f_{2} (x))} \cdot \frac{d (f_{2} (y), f_{2} (x))}{d (y, x)} \\ = & lim_{\begin{matrix} x \to x_{0} \\ y \to x_{0} \\ x \neq y \end{matrix}} \frac{d (f_{1} (f_{2} (y)), f_{1} (f_{2} (x)))}{d (f_{2} (y), f_{2} (x))} \cdot lim_{\begin{matrix} x \to x_{0} \\ y \to x_{0} \\ x \neq y \end{matrix}} \frac{d (f_{2} (y), f_{2} (x))}{d (y, x)} \\ = & a_{1} (f_{2} (x_{0})) \cdot a_{2} (x_{0}) . \end{matrix}

□

Corollary 4.

f_{i}

i = 1, \dots, N

, are continuous strong conformal mappings with strong dilatations

a_{i}

, then the composition

f_{1} \circ f_{2} \circ \dots \circ f_{N}

is strong conformal with strong dilatation

a = (a_{1} \circ f_{2} \circ \dots \circ f_{N}) \cdot (a_{2} \circ f_{3} \circ \dots \circ f_{N}) \dots (a_{N - 1} \circ f_{N}) \cdot a_{N} .

Equip the Cartesian product

X \times X

with the product metric

\tilde{d}

defined by

\tilde{d} ((x_{1}, y_{1}), (x_{2}, y_{2})) = d (x_{1}, x_{2}) + d (y_{1}, y_{2}) .

The diagonal is defined as

Δ = {(x, x) : x \in X} \subset X \times X

. The diagonal is a closed subset of

X \times X

, and, in particular,

Δ

is compact if X is compact. By a neighborhood of the diagonal

Δ

, we mean any open subset

U \supset Δ

X \times X

Lemma 4.

Assume that

(X, d)

is a compact metric space. Let

U

be a neighborhood of the diagonal

Δ \subset X \times X

. There exists

δ = δ_{U} > 0

such that for any

x, y \in X

(x, y) \in U whenever d (x, y) < δ .

Proof.

Due to the compactness of the diagonal

Δ

, there exists

δ > 0

such that

(x, y) \in U

whenever

\tilde{d} ((x, y), (x, x)) < δ

. From the definition of

\tilde{d}

, we have

\tilde{d} ((x, y), (x, x)) = d (x, x) + d (x, y) = 0 + d (x, y) = d (x, y) .

This completes the proof. □

Proposition 2.

Assume that

f : X \to X

is a strong regular conformal mapping with a strong dilatation a. The function

{\tilde{a}}_{f} : X \times X \to R

{\tilde{a}}_{f} (x, y) = \{\begin{matrix} \frac{d (f (x), f (y)}{d (x, y)}, & when x \neq y \\ a (x), & when x = y, \end{matrix}

is continuous. In particular,

{\tilde{a}}_{f}

is uniformly continuous when X is compact. Moreover, if X is compact,

{\tilde{a}}_{f}

is positive in some neighborhood of the diagonal

Δ \subset X \times X

Proof.

The continuity of f implies the continuity of the function

{\tilde{a}}_{f}

at every point

(x, y) \in X \times X

where

x \neq y

. Let

x_{0} \in X

and

ε > 0

. From the conformality condition, there exists

δ_{1} > 0

such that

| {\tilde{a}}_{f} (x, y) - a (x_{0}) | < ε

, when

x \neq y

d (x, x_{0}) < δ_{1}

, and

d (y, x_{0}) < δ_{1}

. Since a is continuous, there exists

δ_{2} > 0

such that

| {\tilde{a}}_{f} (x, x) - a (x_{0}) | = | a (x) - a (x_{0}) | < ε

, when

d (x, x_{0}) < δ_{2}

. Consequently, for

δ = min {δ_{1}, δ_{2}}

| {\tilde{a}}_{f} (x, y) - a (x_{0}) | < ε, when d (x, x_{0}) < δ, d (y, x_{0}) < δ .

The definition of the metric

\tilde{d}

yields

| {\tilde{a}}_{f} (x, y) - a (x_{0}) | < ε, when \tilde{d} ((x, y), (x_{0}, y_{0})) < δ .

This proves the continuity of the function

{\tilde{a}}_{f}

. Since

a > 0

{\tilde{a}}_{f}

is positive in some neighborhood of

Δ

. □

3. Dynamical Solenoids

Let

(X, d)

be a (compact) metric space. For any natural number

n \geq 0

, let

X_{n} = X

. Consider

X_{\infty} = {(X_{n})}_{n \geq 0}

, and let

f_{\infty}

denote the sequence of continuous surjections

f_{n} : X_{n} \to X_{n - 1}

for

n \geq 1

. The system

(X_{\infty}, f_{\infty})

is called a dynamical solenoid. If all elements of the sequence

f_{\infty}

are strong conformal mappings, we place

a_{\infty} = {(a_{n})}_{n \geq 1},

where

a_{n}

denotes the strong dilatation of

f_{n}

. If the sequence

f_{\infty}

consists of strong regular conformal mappings, then

(X_{\infty}, f_{\infty}, a_{\infty})

is called a regular dynamical conformal solenoid.

For any integers

0 \leq i < j

, define

f_{i, j} = f_{i + 1} \circ \dots \circ f_{j} : X_{j} ⟶ X_{i} .

Clearly,

f_{j - 1, j} = f_{j} : X_{j} \to X_{j - 1}

. Also, for all integers

j \geq 0

, let

f_{j, j} = {id}_{X_{j}}

. It follows from the definitions that for any integers

0 \leq i \leq k \leq j

f_{i, k} \circ f_{k, j} = f_{i, j} .

In the case when

(X_{\infty}, f_{\infty}, a_{\infty})

is a strong regular dynamical conformal solenoid, by Corollary 4, for

0 \leq i < j

, each

f_{i, j}

is a strong regular conformal mapping with strong dilatation

\begin{matrix} a_{i, j} & = & a_{i + 1} \circ f_{i + 1, j} \cdot a_{i + 2} \circ f_{i + 2, j} \dots a_{j - 1} \circ f_{j - 1, j} \cdot a_{j} \circ f_{j, j} \\ = & \prod_{k = 1}^{j - i} a_{i + k} \circ f_{i + k, j} . \end{matrix}

Moreover, we can set

a_{i, i} = 1

, which is consistent with the fact that

f_{i, i}

is the identity mapping on

X_{i}

, and, thus, conformal with unit dilatation.

For

x \in X

n \geq 0

, and

δ > 0

, the n-th dynamical ball

B (x, n, δ) \subset X_{n} = X

is defined as follows

\begin{matrix} B (x, n, δ) & = & {y \in X_{n} : d (f_{i, n} (x), f_{i, n} (y)) < δ, \forall_{1 \leq i \leq n}} \\ = & ⋂_{i = 0}^{n} {f_{i, n}}^{- 1} (B (f_{i, n} (x), δ)) . \end{matrix}

Since,

{f_{n, n}}^{- 1} (B (f_{n, n} (x), δ)) = B (x, δ),

for any

n \geq 0

, we have

B (x, n, δ) \subset B (x, δ) and B (x, 0, δ) = B (x, δ) .

(2)

Let us assume that a sequence of continuous functions

φ_{\infty} = {(φ_{n} : X_{n} \to R)}_{n \geq 1}

is given, which we will call a multipotential. For

n \geq 1

, we define

\begin{matrix} S_{n}^{f_{\infty}} φ_{\infty} & = & \sum_{i = 1}^{n} φ_{i} \circ f_{i, n} : X_{n} \to R . \end{matrix}

3.1. Lyapunov Exponents for Dynamical Solenoids

In the case when

(X_{\infty}, f_{\infty}, a_{\infty})

is a regular dynamical conformal solenoid, for

n \geq 1

, we set

\begin{matrix} λ_{n}^{f_{\infty}} & = & log (\sqrt[n]{a_{0, n}}) = \frac{1}{n} log (\prod_{k = 1}^{n} a_{k} \circ f_{k, n}) \\ = & \frac{1}{n} \sum_{k = 1}^{n} (log a_{k}) \circ f_{k, n} = \frac{1}{n} S_{n}^{f_{\infty}} (log a_{\infty}) . \end{matrix}

More precisely, for

x \in X_{n}

, the function

λ_{n}^{f_{\infty}} : X_{n} \to R

, is defined by

λ_{n}^{f_{\infty}} (x) = \frac{1}{n} S_{n}^{f_{\infty}} (log a_{\infty}) (x) = \frac{1}{n} \sum_{k = 1}^{n} log a_{k} (f_{k, n} (x)) .

For

1 \leq i \leq j

, we also define

\begin{matrix} S_{i, j}^{f_{\infty}} (log a_{\infty}) (x) & = & \sum_{k = i}^{j} log a_{k} (f_{k, j} (x)) \\ λ_{i, j}^{f_{\infty}} (x) & = & \frac{1}{j - i + 1} S_{i, j}^{f_{\infty}} (log a_{\infty}) (x) . \end{matrix}

Note that

\begin{matrix} S_{j, j}^{f_{\infty}} (log a_{\infty}) & = & log a_{j}, S_{1, n}^{f_{\infty}} = S_{n}^{f_{\infty}}, \\ λ_{j, j}^{f_{\infty}} & = & log a_{j}, λ_{1, n}^{f_{\infty}} = λ_{n}^{f_{\infty}} . \end{matrix}

Let

x \in X

. Without any additional assumptions, there is no reason for the sequence

(λ_{n}^{f_{\infty}} (x))

to converge. However, there exist the upper limit

\bar{λ^{f_{\infty}}} (x)

and the lower limit

\underset{̲}{λ^{f_{\infty}}} (x)

, called, respectively, the upper and lower Lyapunov exponents. More precisely,

\bar{λ^{f_{\infty}}} (x) = \underset{n \to \infty}{lim sup} λ_{n}^{f_{\infty}} (x), \underset{̲}{λ^{f_{\infty}}} (x) = \underset{n \to \infty}{lim inf} λ_{n}^{f_{\infty}} (x) .

In the case where the sequence

(λ_{n}^{f_{\infty}} (x))

converges, its limit is called the Lyapunov exponent and is denoted by

λ^{f_{\infty}} (x)

, i.e.,

λ^{f_{\infty}} (x) = lim_{n \to \infty} λ_{n}^{f_{\infty}} (x) .

For any

E \subset R

, let

A (E) = {x \in X : [\underset{̲}{λ^{f_{\infty}}} (x), \bar{λ^{f_{\infty}}} (x)] \subset E}

In particular, when

E = {α}

, we write

A (α)

. In this case,

A (α) = {x \in X : λ^{f_{\infty}} (x) = α} .

3.2. Topological Pressure for Dynamical Solenoids

Let

Z \subset X

. For each natural number

N \geq 1

and each real number

δ > 0

, consider the family

C (Z, N, δ)

of all dynamical balls

B (x, n, δ)

such that

x \in Z

and

n \geq N

. Since X is compact, there is a finite or countable set of balls

{B (x_{i}, n_{i}, δ)} \subset C (Z, N, δ)

indexed by pairs

(x_{i}, n_{i})

, which forms an open cover of the set Z. We denote the family of all such countable covers by

P (Z, N, δ)

. We denote the family of all sets

{(x_{i}, n_{i})}

indexing these covers by

I (Z, N, δ)

, i.e.,

{(x_{i}, n_{i})} \in I (Z, N, δ) \Leftrightarrow {B (x_{i}, n_{i}, δ)} \in P (Z, N, δ) .

Let

s \in R

. For

I = {(x_{i}, n_{i})} \in I (Z, N, δ)

and

t \in R

, we define

\begin{matrix} Φ (s, I) = Φ (s, I, φ_{\infty}) = \sum_{(x_{i}, n_{i}) \in I} exp (- n_{i} s + (S_{n_{i}}^{f_{\infty}} φ_{\infty}) (x_{i})), \\ Φ (s, I, t) = Φ (s, I, t, φ_{\infty}) = \sum_{(x_{i}, n_{i}) \in I} exp (- n_{i} s + t (S_{n_{i}}^{f_{\infty}} φ_{\infty}) (x_{i})) . \end{matrix}

Note that

Φ (s, I), Φ (s, I, t) \in (0, \infty]

and

Φ (s, I) = Φ (s, I, 1)

. Directly from the definition of the operator

S_{n}^{f_{\infty}}

, it follows that

Φ (s, I, t φ_{\infty}) = Φ (s, I, t, φ_{\infty}) .

Note that every cover belonging to the family

P (Z, N + 1, δ)

, belongs to

P (Z, N, δ)

, i.e.,

P (Z, N + 1, δ) \subset P (Z, N, δ) .

(3)

As a consequence, we obtain the inclusion

I (Z, N + 1, δ) \subset I (Z, N, δ) .

(4)

We define

\begin{matrix} m_{P} (Z, s, φ_{\infty}, N, δ) & = & inf \{Φ (s, I, φ_{\infty}) : I \in I (Z, N, δ)\}, \end{matrix}

which has values in

[0, \infty] .

Lemma 5.

The function

N \mapsto m_{P} (Z, s, φ_{\infty}, N, δ), N \in {1, 2, 3, \dots}

is nondecreasing.

Proof.

A direct consequence of the inclusion (4) and the properties of the lower bound. □

By Lemma 5, it follows that there exists a limit

m_{P} (Z, s, φ_{\infty}, δ) = lim_{N \to \infty} m_{P} (Z, s, φ_{\infty}, N, δ) \in [0, \infty] .

This limit is equal to

sup \{m_{P} (Z, s, φ_{\infty}, N, δ) : N \geq 1\} \in [0, \infty] .

Lemma 6.

The functions

\begin{matrix} s & \mapsto & m_{P} (Z, s, φ_{\infty}, N, δ), \\ s & \mapsto & m_{P} (Z, s, φ_{\infty}, δ), w h e r e s \in R, \end{matrix}

are nonincreasing.

Proof.

Let

t \geq s

. Then, for any integer

n \geq 1

e^{- n s} \geq e^{- n t}

. Therefore, using the definition of

Φ (s, I)

, we have

\begin{matrix} Φ (s, I) & = \sum_{(x_{i}, n_{i}) \in I} exp (- n_{i} s + (S_{n_{i}}^{f_{\infty}} φ_{\infty}) (x_{i})) \\ = \sum_{(x_{i}, n_{i}) \in I} e^{- n_{i} s} exp ((S_{n_{i}}^{f_{\infty}} φ_{\infty}) (x_{i})) \\ \geq \sum_{(x_{i}, n_{i}) \in I} e^{- n_{i} t} exp ((S_{n_{i}}^{f_{\infty}} φ_{\infty}) (x_{i})) \\ = \sum_{(x_{i}, n_{i}) \in I} exp (- n_{i} t + (S_{n_{i}}^{f_{\infty}} φ_{\infty}) (x_{i})) = Φ (t, I) . \end{matrix}

Therefore, we have

\begin{matrix} Φ (s, I) & \geq & Φ (t, I) \geq inf \{Φ (t, I) : I \in I (Z, N, δ)\} \\ = & m_{P} (Z, t, φ_{\infty}, N, δ) . \end{matrix}

It follows that

m_{P} (Z, s, φ_{\infty}, N, δ) \geq m_{P} (Z, t, φ_{\infty}, N, δ), for t \geq s,

which proves that the first of the functions in the formulation of the lemma is nonincreasing. Next, by passing to the limit in the above inequality with

N \to \infty

, we obtain

m_{P} (Z, s, φ_{\infty}, δ) \geq m_{P} (Z, t, φ_{\infty}, δ), for t \geq s,

which proves that the second of the functions in the formulation of the lemma is nonincreasing.

□

Lemma 7.

The functions

\begin{matrix} δ & \mapsto & m_{P} (Z, s, φ_{\infty}, N, δ), \\ δ & \mapsto & m_{P} (Z, s, φ_{\infty}, δ), w h e r e δ > 0, \end{matrix}

are nonincreasing.

Proof.

Let

η \geq δ > 0

. Then,

B (x, n, δ) \subset B (x, n, η)

. Hence,

I (Z, N, δ) \subset I (Z, N, η)

and, therefore,

\begin{matrix} m_{P} (Z, s, φ_{\infty}, N, δ) & = & inf {Φ (s, I) : I \in I (Z, N, δ)} \\ \geq & inf {Φ (s, I) : I \in I (Z, N, η)} \\ = & m_{P} (Z, s, φ_{\infty}, N, η), \end{matrix}

which proves the monotonicity of the first function in the lemma. Next, by passing to the limit in the above inequality with

N \to \infty

, we obtain the monotonicity of the second function. □

Corollary 5.

(a): The function

$(s, N, δ) \mapsto m_{P} (Z, s, φ_{\infty}, N, δ),$

where $(s, N, δ) \in R \times {1, 2, \dots} \times (0, \infty)$ , is nonincreasing with respect to s and δ, and nondecreasing with respect to N.
(b): The function

$(s, δ) \mapsto m_{P} (Z, s, φ_{\infty}, δ),$

where $(s, δ) \in R \times (0, \infty)$ , is nonincreasing with respect to s and δ.

Lemma 8.

Let

t > s

. Then,

(a): If $m_{P} (Z, s, φ_{\infty}, δ) < \infty$ , then $m_{P} (Z, t, φ_{\infty}, δ) = 0$ .
(b): If $m_{P} (Z, t, φ_{\infty}, δ) > 0$ , then $m_{P} (Z, s, φ_{\infty}, δ) = \infty$ .

Proof.

Notice that for

I \in I (Z, N, δ)

Φ (t, I) \leq e^{- N (t - s)} \cdot Φ (s, I),

(5)

or, equivalently,

Φ (s, I) \geq e^{N (t - s)} \cdot Φ (t, I) .

(6)

(a) Using (5), we obtain the following:

\begin{matrix} m_{P} (Z, t, φ_{\infty}, N, δ) & = inf \{Φ (t, I) : I \in I (Z, N, δ)\} \\ \leq Φ (t, I) \leq e^{- N (t - s)} \cdot Φ (s, I), \\ m_{P} (Z, t, φ_{\infty}, N, δ) & \leq e^{- N (t - s)} \cdot inf \{Φ (s, I) : I \in I (Z, N, δ)\} \\ = e^{- N (t - s)} \cdot m_{P} (Z, s, φ_{\infty}, N, δ) \\ \leq e^{- N (t - s)} \cdot sup {m_{P} (Z, s, φ_{\infty}, N, δ) : N \geq 1} \\ = e^{- N (t - s)} \cdot m_{P} (Z, s, φ_{\infty}, δ) \end{matrix}

Since

t - s > 0

{lim}_{N \to \infty} e^{- N (t - s)} = 0

. Consequently, assuming that

m_{P} (Z, s, φ_{\infty}, δ) < \infty

, we obtain

\begin{matrix} m_{P} (Z, t, φ_{\infty}, δ) & = lim_{N \to \infty} m_{P} (Z, t, φ_{\infty}, N, δ) \\ \leq m_{P} (Z, s, φ_{\infty}, δ) \cdot lim_{N \to \infty} e^{- N (t - s)} = 0, \end{matrix}

which proves (a).

The proof of (b) is analogous to the proof of item (a). □

The function

s \mapsto m_{P} (Z, s, φ_{\infty}, δ)

has a unique critical point which is denoted by

P_{Z} (φ_{\infty}, δ)

(by Lemma 8).

More precisely,

\begin{matrix} P_{Z} (φ_{\infty}, δ) & = & sup {s \in R : m_{P} (Z, s, φ_{\infty}, δ) = \infty} \\ = & inf {s \in R : m_{P} (Z, s, φ_{\infty}, δ) = 0} . \end{matrix}

(7)

Thus, by Lemma 7, it follows that there exists a limit:

P_{Z} (φ_{\infty}) = lim_{δ \to 0^{+}} P_{Z} (φ_{\infty}, δ) = sup {P_{Z} (φ_{\infty}, δ) : δ > 0} .

(8)

The limit

P_{Z} (φ_{\infty})

is called the topological pressure of the dynamical solenoid with respect to the subset

Z \subset X

and the multipotential

φ_{\infty}

Lemma 9.

Z_{1} \subset Z_{2} \subset X

, then

P_{Z_{1}} (φ_{\infty}) \leq P_{Z_{2}} (φ_{\infty}) .

Proof.

It easily follows from the definition and the properties stated above. □

As a corollary, we obtain the following lemma.

Lemma 10.

{(Z_{m})}_{m \in N}

is a sequence of subsets of the space X such that

⋃_{m \in N} Z_{m} = Z

and

Z_{m} \subset Z_{m + 1}

, then

P_{Z} (φ_{\infty}) = sup_{m \in N} P_{Z_{m}} (φ_{\infty}) = lim_{m \to \infty} P_{Z_{m}} (φ_{\infty}) .

3.3. Topological Entropy and Pressure for Dynamical Solenoids

Recall the definition of topological entropy

h_{t o p} (f_{\infty}, Z)

restricted to a set

Z \subset X

. Define

m_{h}^{f_{\infty}} (Z, t, N, δ) = inf \{\sum_{\begin{matrix} (x_{i}, n_{i}) \in I \end{matrix}} exp (- n_{i} \cdot t) : : Z \subset ⋃_{\begin{matrix} (x_{i}, n_{i}) \in I \end{matrix}} B (x_{i}, n_{i}, δ), n_{i} \geq N, x_{i} \in Z\} .

Then, we have the following relations:

\begin{matrix} m_{h}^{f_{\infty}} (Z, t, δ) & = & lim_{N \to \infty} m_{h}^{f_{\infty}} (Z, t, N, δ), \\ h_{Z} (f_{\infty}, δ) & = & inf {t : m_{h}^{f_{\infty}} (Z, t, δ) = 0}, \\ h_{t o p} (f_{\infty}, Z) & = & lim_{δ \to 0} h_{Z} (f_{\infty}, δ) . \end{matrix}

The quantity

h_{t o p} (f_{\infty}, Z)

is called the topological entropy of the dynamical solenoid

(X_{\infty}, f_{\infty})

restricted to the set Z.

Let

0_{\infty}

denote the multipotential which is the sequence of zero functions. By previous definitions, we immediately obtain

m_{h}^{f_{\infty}} (Z, t, N, δ) = m_{P} (Z, t, 0_{\infty}, N, δ) .

Consequently,

P_{Z} (0_{\infty}) = h_{t o p} (f_{\infty}, Z) .

(9)

Proposition 3.

Let

α, β \in R

and

α \leq β

. Assume that for any

x \in Z

α \leq \underset{n \to \infty}{lim inf} \frac{1}{n} (S_{n}^{f_{\infty}} φ_{\infty}) (x) \leq \underset{n \to \infty}{lim sup} \frac{1}{n} (S_{n}^{f_{\infty}} φ_{\infty}) (x) \leq β .

(10)

Then

(a): For $h > 0$ and $t \in R$ ,

$P_{Z} (t φ_{\infty}) + α h \leq P_{Z} ((t + h) φ_{\infty}) \leq P_{Z} (t φ_{\infty}) + β h .$
(b): For $h < 0$ and $t \in R$ ,

$P_{Z} (t φ_{\infty}) + β h \leq P_{Z} ((t + h) φ_{\infty}) \leq P_{Z} (t φ_{\infty}) + α h .$

Proof.

Let

ε > 0

. For any natural

m \geq 1

, let

Z_{m} = ⋂_{n \geq m} \{z \in Z : α - ε < \frac{1}{n} (S_{n}^{f_{\infty}} φ_{\infty}) (z) < β + ε\} .

Directly, for the definition,

{(Z_{m})}_{m \geq 1}

is an increasing sequence of sets, i.e.,

Z_{m} \subset Z_{m + 1}

. Moreover,

Z = ⋃_{m \geq 1} Z_{m}

. The inclusion “⊃" follows directly from the definition of sets

Z_{m}

. However, the second inclusion “⊂" follows from the properties of the limit inferior and limit superior of the sequence.

This implies that, by Lemma 10,

P_{Z} (φ_{\infty}) = sup {P_{Z_{m}} (φ_{\infty}) : m \geq 1} .

(11)

(a) Fix

t \in R

and

h > 0

. For

N \geq m

and

I \in I (Z_{m}, N, δ)

we have the following estimations:

\begin{matrix} Φ (s, I, (t + h) φ_{\infty}) & = Φ (s, I, (t + h), φ_{\infty}) \geq Φ (s - h (α - ε), t φ_{\infty}), \\ Φ (s, I, (t + h) φ_{\infty}) & = Φ (s, I, (t + h), φ_{\infty}) \leq Φ (s - h (β + ε), t φ_{\infty}) . \end{matrix}

This implies the estimations

\begin{matrix} m_{P} (Z_{m}, s - h (α - ε), t φ_{\infty}, N, δ) & \leq m_{P} (Z_{m}, s, (t + h) φ_{\infty}, N, δ) \\ \leq & m_{P} (Z_{m}, s - h (β + ε), t φ_{\infty}, N, δ), \\ m_{P} (Z_{m}, s - h (α - ε), t φ_{\infty}, δ) \leq & m_{P} (Z_{m}, s, (t + h) φ_{\infty}, δ) \\ \leq & m_{P} (Z_{m}, s - h (β + ε), t φ_{\infty}, δ) . \end{matrix}

If the points

s^{'}

and

s^{″}

coincide with critical points of functions

\begin{matrix} s & \mapsto m_{P} (Z_{m}, s - h (α - ε), t φ_{\infty}, δ), \\ s & \mapsto m_{P} (Z_{m}, s - h (β + ε), t φ_{\infty}, δ), \end{matrix}

then

\begin{matrix} s^{'} \leq P_{Z_{m}} ((t + h) φ_{\infty}, δ), & s^{'} - h (α - ε) = P_{Z_{m}} (t φ_{\infty}, δ), \\ s^{″} \geq P_{Z_{m}} ((t + h) φ_{\infty}, δ), & s^{″} - h (β + ε) = P_{Z_{m}} (t φ_{\infty}, δ), \end{matrix}

which leads to the following estimations:

\begin{matrix} P_{Z_{m}} (t φ_{\infty}, δ) + h (α - ε) & \leq P_{Z_{m}} ((t + h) φ_{\infty}, δ) \\ \leq P_{Z_{m}} (t φ_{\infty}, δ) + h (β + ε), \\ P_{Z_{m}} (t φ_{\infty}) + h (α - ε) & \leq P_{Z_{m}} ((t + h) φ_{\infty}) \\ \leq P_{Z_{m}} (t φ_{\infty}) + h (β + ε) . \end{matrix}

Hence, by (11), passing to the limit with

ε \to 0

, we obtain (a).

The proof of (b) is analogous to the proof of item (a). □

Remark 1.

The estimation in Proposition 3(b) can be obtained by (a) by taking

t - h

instead of t. We can do this because t is an arbitrary real number. Then, for

h > 0

, we obtain

P_{Z} ((t - h) φ_{\infty}) + α h \leq P_{Z} (t φ_{\infty}) \leq P_{Z} ((t - h) φ_{\infty}) + β h,

which is equivalent to the estimation

\begin{matrix} P_{Z} (t φ_{\infty}) + β (- h) & \leq P_{Z} ((t + (- h)) φ_{\infty}) \\ \leq P_{Z} (t φ_{\infty}) + α (- h), h > 0 . \end{matrix}

(12)

Inputting (12)

- h

instead of h, we obtain Proposition 3(b).

Notice that

\begin{matrix} \underset{n \to \infty}{lim inf} \frac{1}{n} S_{n}^{f_{\infty}} (- log a_{\infty}) & = \underset{n \to \infty}{lim inf} \frac{1}{n} (- S_{n}^{f_{\infty}} (log a_{\infty})) \\ = - \underset{n \to \infty}{lim sup} \frac{1}{n} S_{n}^{f_{\infty}} (log a_{\infty}) = - \bar{λ^{f_{\infty}}}, \\ \underset{n \to \infty}{lim sup} \frac{1}{n} S_{n}^{f_{\infty}} (- log a_{\infty}) & = \underset{n \to \infty}{lim sup} \frac{1}{n} (- S_{n}^{f_{\infty}} (log a_{\infty})) \\ = - \underset{n \to \infty}{lim inf} \frac{1}{n} S_{n}^{f_{\infty}} (log a_{\infty}) = - \underset{̲}{λ^{f_{\infty}}} . \end{matrix}

Therefore, taking

φ_{\infty} = - log a_{\infty}

in Proposition 3, we obtain that the condition (10) is equivalent to the following:

- β \leq \underset{̲}{λ^{f_{\infty}}} (x) \leq \bar{λ^{f_{\infty}}} (x) \leq - α, x \in Z .

(13)

Corollary 6.

Let

c, C \in R

with

c \leq C

. Assume that for any

x \in Z

c \leq \underset{̲}{λ^{f_{\infty}}} (x) \leq \bar{λ^{f_{\infty}}} (x) \leq C .

Then

(a): For $h > 0$ and $t \in R$ ,

$P_{Z} (- t log a_{\infty}) - C h \leq P_{Z} (- (t + h) log a_{\infty}) \leq P_{Z} (- t log a_{\infty}) - c h .$
(b): For $h < 0$ and $t \in R$ ,

$P_{Z} (- t log a_{\infty}) - c h \leq P_{Z} (- (t + h) log a_{\infty}) \leq P_{Z} (- t log a_{\infty}) - C h .$
(c): If $c > 0$ , the pressure function $t \mapsto P_{Z} (- t log a_{\infty})$ is a continuous, strictly decreasing function with Lipschitz constant C.
(d): If $C < 0$ , the pressure function $t \mapsto P_{Z} (- t log a_{\infty})$ is a continuous, strictly increasing function with Lipschitz constant $| c |$ .
(e): If $c = C$ , i.e., the Lyapunov exponent $λ^{f_{\infty}}$ is constant on Z, then the pressure function $t \mapsto P_{Z} (- t log a_{\infty})$ reduces to a linear function of form $t \mapsto h_{t o p} (f_{\infty}, Z) - λ^{f_{\infty}} \cdot t$ . In particular, if $c = C = 0$ , then the pressure function is constant and equal to $h_{t o p} (f_{\infty}, Z)$ .

Proof.

Estimations (a) and (b) are a direct consequence of the condition (13) and Proposition 3; where we take take

α = - C

and

β = - c

. (c) Since

c > 0

, for

h > 0

we obtain

c h > 0

. Hence, by (a),

P_{Z} (- (t + h) log a_{\infty}) \leq P_{Z} (- t log a_{\infty}) - c h < P_{Z} (- t log a_{\infty}),

which implies that the function

t \mapsto P_{Z} (- t log a_{\infty})

is strictly decreasing. Thus, by (a), we obtain

0 < P_{Z} (- t log a_{\infty}) - P_{Z} (- (t + h) log a_{\infty}) \leq C h, t > 0 .

Consequently,

| P_{Z} (- t_{1} log a_{\infty}) - P_{Z} (- t_{2} log a_{\infty}) | \leq C | t_{1} - t_{2} |, t_{1}, t_{2} \in R .

Therefore, the function

t \mapsto P_{Z} (- t log a_{\infty})

is a continuous with Lipschitz constant C.

(d) Since

C < 0

, for

h > 0

- C h = | C | h > 0

. Hence, by (a),

P_{Z} (- (t + h) log a_{\infty}) \geq P_{Z} (- t log a_{\infty}) + | C | h > P_{Z} (- t log a_{\infty}),

which implies that the function

t \mapsto P_{Z} (- t log a_{\infty})

is strictly increasing. Thus, by (a), we obtain

0 < P_{Z} (- (t + h) log a_{\infty}) - P_{Z} (- t log a_{\infty}) \leq - c h = | c | h, t > 0 .

Consequently,

| P_{Z} (- t_{1} log a_{\infty}) - P_{Z} (- t_{2} log a_{\infty}) | \leq | c | | t_{1} - t_{2} |, t_{1}, t_{2} \in R,

which yields the Lipschitz continuity of the function.

(e) Observe first that

P_{Z} (0 log a_{\infty}) = P_{Z} (0_{\infty}) = h_{t o p} (f_{\infty}, Z)

, by (9). The equality

c = C

implies that

c = C = λ^{f_{\infty}}

and

λ^{f_{\infty}}

is constant on Z. Combining (a) and (b), we conclude that for any

t, h \in R

P_{Z} (- t log a_{\infty}) - λ^{f_{\infty}} h = P_{Z} (- (t + h) log a_{\infty}) .

Substituting

h = - t

we obtain

P_{Z} (- t log a_{\infty}) + λ^{f_{\infty}} t = P_{Z} (- 0 log a_{\infty}) = P_{Z} (0_{\infty}) = h_{t o p} (f_{\infty}, Z) .

Hence,

P_{Z} (- t log a_{\infty}) = h_{t o p} (f_{\infty}, Z) - λ^{f_{\infty}} t

. □

3.4. Proof of Theorem 3

Proof.

Let

h_{1} = \frac{h_{t o p} (f_{\infty}, Z)}{C}, h_{2} = \frac{h_{t o p} (f_{\infty}, Z)}{c} .

Observe first that

h_{1} \leq h_{2}

whenever

0 < c \leq C

c \leq C < 0

. We know by Corollary 6(c) and (d) that the pressure function

t \mapsto P_{Z} (- t log a_{\infty})

is continuous and strictly increasing for

C < 0,

while it is strictly decreasing for

c > 0

. In both cases, it follows that there exists at most one root

t^{'} \in [h_{1}, h_{2}]

such that

P_{Z} (- t^{'} log a_{\infty}) = 0

Consequently, to prove the assertion, it suffices to show that the pressure function has different signs at the ends of the interval

[h_{1}, h_{2}]

, or, equivalently,

P_{Z} (- h_{1} log a_{\infty}) P_{Z} (- h_{2} log a_{\infty}) \leq 0 .

(14)

0 < c \leq C

, then

h_{1} \leq h_{2}

. Moreover, by Corollary 6(a) with

t = 0

, we obtain

P_{Z} (- h_{1} log a_{\infty}) \geq 0, P_{Z} (- h_{2} log a_{\infty}) \leq 0;

thus, (14) follows.

c \leq C < 0

, then

h_{1} \leq h_{2}

. Moreover, by Corollary 6(b) with

t = 0

, we obtain

P_{Z} (- h_{1} log a_{\infty}) \leq 0, P_{Z} (- h_{2} log a_{\infty}) \geq 0;

thus, (14) follows. The proof is complete. □

4. Hausdorff Dimension

Let us recall the definition of the Hausdorff measure and dimension. Let

(Y, d)

be a metric space and

Z \subset Y

. Let

δ > 0

. A family

V

of subsets of the space Y is called a countable δ-cover of the set Z if

(1): The family $V$ consists of at most countably many sets.
(2): The family $V$ covers the set Z, i.e., $Z \subset ⋃ V$ .
(3): The diameter of each set in the family $V$ is $< δ$ , i.e., for any $V \in V$ , $diam V < δ$ .

The family of all countable

δ

-covers of the set Z is denoted by

D (Z, δ)

. For real numbers

s \geq 0

and

δ > 0

, let us define

H^{s} (Z, δ) = inf \{\sum_{V \in V} {[diam (V)]}^{s} : V \in D (Z, δ)\} .

The mapping

δ \mapsto H^{s} (Z, δ)

is monotonically increasing with respect to

δ

. Therefore, there exists a limit

H^{s} (Z) = lim_{δ \to 0} H^{s} (Z, δ) \in [0, + \infty],

which is called the s-dimensional Hausdorff measure of the set Z. The graph of the function

s \mapsto H^{s} (Z)

is symbolically illustrated below.

The function has exactly one critical point

s_{0}

where it “drops” from infinity to zero. The critical point

s_{0}

is called the Hausdorff dimension of the set Z and is denoted by

{dim}_{H} (Z)

, i.e.,

\begin{matrix} {dim}_{H} (Z) & = inf \{s \geq 0 : H^{s} (Z) = 0\} \\ = sup \{s \geq 0 : H^{s} (Z) = + \infty\} . \end{matrix}

One may equivalently define Hausdorff measure and Hausdorff dimension using covers by open balls; to this definition, we will refer in the proof of Theorem 4. For a subset

Z \subset Y

, let us define

\begin{matrix} M_{H} (Z, t, r) & = inf \{\sum_{(x_{i}, r_{i}) \in I} {(2 \cdot r_{i})}^{t} : Z \subset ⋃_{(x_{i}, r_{i}) \in I} B (x_{i}, r_{i}), r_{i} < r, x_{i} \in Z\}, \end{matrix}

where I is a set of at most countably many elements. The mapping

r \mapsto M_{H} (Z, t, r)

is monotonic with respect to r. Therefore, there exists a limit

M_{H} (Z, t) = lim_{r \to 0} M_{H} (Z, t, r) \in [0, + \infty] .

Varying

t > 0

, we obtain a monotonic function

t \mapsto M_{H} (Z, t)

. This function has exactly one critical point, denoted by

{dim}_{H}^{*} (Z)

, where it “drops" from infinity to zero, i.e.,

\begin{matrix} {dim}_{H}^{*} (Z) & = & inf \{t > 0 : M_{H} (Z, t) = 0\} \\ = & sup \{t > 0 : M_{H} (Z, t) = + \infty\} . \end{matrix}

Proposition 4.

For any subset Z of a metric space

(Y, d)

, the following equality holds:

{dim}_{H} (Z) = {dim}_{H}^{*} (Z) .

Proof.

See [12] the proof of Proposition 5.1. □

5. Properties of Dynamical Balls of DCSE-Solenoids

We say that a regular dynamical conformal solenoid

(X_{\infty}, f_{\infty}, a_{\infty})

is simple and expanding (in short: DCSE-solenoid) if

(1): Each map $f_{n}$ is a strong conformal map with strong dilatation $a_{n}$ .
(2): There exists $K > 1$ such that for any $n \geq 1$ and any $x \in X$ , $1 < a_{n} (x) < K$ .
(3): The sequence $f_{\infty}$ consists of only finitely many distinct maps.

We will show that Bowen’s formula holds for DCSE-solenoids.

Lemma 11.

Let f be a strong regular conformal map with strong dilatation

a > 0

defined on a compact metric space

(X, d)

. We define

A_{f} (x, y) = \{\begin{matrix} log d (f (x), f (y)) - log d (x, y), & when x \neq y \\ log a (x), & when x = y . \end{matrix}

Then, there exists a neighborhood

U_{f}

of the diagonal

Δ \subset X \times X

, such that

A_{f}

is well defined and uniformly continuous on

U_{f}

Proof.

A direct consequence of Proposition 2. □

Corollary 7.

Assume that

(X_{\infty}, f_{\infty}, a_{\infty})

is a DCSE-solenoid. Then,

(a): There exists a neighborhood $U$ of the diagonal $Δ \subset X \times X$ , such that the sequence $(A_{f_{n}})$ is well defined and uniformly continuous in $U$ .
(b): For any $ε > 0$ , there exists $δ^{'} > 0$ such that for any natural number $n \geq 1,$

$d (f_{n} (x), f_{n} (y)) e^{- (log a_{n} (x) + ε)} < d (x, y) < d (f_{n} (x), f_{n} (y)) e^{- (log a_{n} (x) - ε)},$

whenever $x, y \in X$ and $d (x, y) < δ^{'}$ .

Proof.

(a) By the definition of a DCSE-solenoid, there exists a natural number

k \geq 1

such that for any natural number n,

f_{n} \in {f_{1}, \dots, f_{k}} .

Let

U_{i} = U_{f_{i}}

be the neighborhood of the diagonal

Δ

, determined by Lemma 11, for each

i = 1, \dots, k

. Set

U = U_{1} \cap U_{2} \cap \dots \cap U_{k}

. In the neighborhood

U

, the sequence

(A_{f_{n}})

is well defined and uniformly continuous since it consists of a finite number of different maps. Thus,

(A_{f_{n}})

is uniformly continuous in

U

(b) Choose

δ_{U} > 0

, determined by Lemma 4, for the neighborhood

U

. Let

ε > 0

. From the equality

\tilde{d} ((x, y), (x, x)) = d (x, y)

(see the proof of Lemma 4) and the uniform continuity of the sequence

(A_{f_{n}})

, there exists a positive

δ^{'} < δ_{U}

such that for any

n \geq 1

| A_{f_{n}} (x, y) - A_{f_{n}} (x, x) | < ε,

(15)

whenever

d (x, y) < δ

. Directly from the definition of the function

A_{f_{n}}

, inequality (15) is equivalent to the double inequality

log d (f_{n} (x), f_{n} (y)) - log a_{n} (x) - ε < log d (x, y) < log d (f_{n} (x), f_{n} (y)) - log a_{n} (x) + ε .

Taking the exponential function exp to all sides of the double inequality, we complete the proof. □

Lemma 12.

Assume that

(X_{\infty}, f_{\infty}, a_{\infty})

is a DCSE-solenoid. Then, for any

ε > 0

, there exists

δ^{'} > 0

such that for any

x \in X

and any natural number

n \geq 1

B (x, δ \cdot exp (- n (λ_{n}^{f_{\infty}} (x) + ε))) \subset B (x, n, δ) \subset B (x, δ \cdot exp (- n (λ_{n}^{f_{\infty}} (x) - ε))),

whenever

0 < δ < δ^{'}

Proof.

Take

ε > 0

. Choose

δ^{'} > 0

determined by Corollary 7 (b). Let

0 < δ < δ^{'}

and

n \geq 1

. For any point

z \in X

, let

z_{i} = f_{i, n} (z)

, for

i = 0, \dots

, n.

In the first step, we intend to prove the inclusion

B (x, n, δ) \subset B (x, δ \cdot e^{- n (λ_{n}^{f_{\infty}} (x) - ε)})

. Consider the dynamical ball

B (x, n, δ) = {y \in X : (d (x_{i}, y_{i}) < δ, i = 0, \dots, n} .

Let

x \in X

and

y \in B (x, n, δ)

. Then

d (x_{i}, y_{i}) < δ

, which means that the second inequality of Corollary 7(b) can be rewritten as follows:

d (x_{i}, y_{i}) < d (f_{i} (x_{i}), f_{i} (y_{i})) e^{- (log a_{i} (x_{i}) - ε)},

i.e., due to the equalities

x_{i - 1} = f_{i} (x_{i})

and

y_{i - 1} = f_{i} (y_{i})

, we obtain

d (x_{i}, y_{i}) < d (x_{i - 1}, y_{i - 1}) e^{- (log a_{i} (x_{i}) - ε)} .

Consequently, using the definition of

λ_{i, j}^{f_{\infty}}

(see Section 3) for

y \in B (x, n, δ)

, we successively obtain the following.

For

n \geq 1

\begin{matrix} d (x, y) & = d (x_{n}, y_{n}) < d (x_{n - 1}, y_{n - 1}) \cdot e^{- (log a_{n} (x_{n}) - ε)} \\ = d (x_{n - 1}, y_{n - 1}) \cdot exp (- (λ_{n, n}^{f_{\infty}} (x) - ε)) . \end{matrix}

For

n \geq 2

\begin{matrix} d (x, y) & = d (x_{n}, y_{n}) < d (x_{n - 1}, y_{n - 1}) \cdot e^{- (log a_{n} (x_{n}) - ε)} \\ < d (x_{n - 2}, y_{n - 2}) \cdot e^{- (log a_{n - 1} (x_{n - 1}) - ε)} \cdot e^{- (log a_{n} (x_{n}) - ε)} \\ = d (x_{n - 2}, y_{n - 2}) \cdot e^{- (log a_{n - 1} (x_{n - 1}) + log a_{n} (x_{n}) - 2 ε)} \\ = d (x_{n - 2}, y_{n - 2}) \cdot exp (- 2 (λ_{n - 1, n}^{f_{\infty}} (x) - ε)) . \end{matrix}

For

n \geq j

d (x, y) < d (x_{n - j}, y_{n - j}) \cdot exp (- j \cdot (λ_{n - j + 1, n}^{f_{\infty}} (x) - ε)) .

Taking

j = n

and applying the equality

λ_{1, n}^{f_{\infty}} = λ_{n}^{f_{\infty}}

we obtain

d (x, y) < d (x_{0}, y_{0}) \cdot exp (- n \cdot (λ_{n}^{f_{\infty}} (x) - ε)) < δ \cdot exp (- n \cdot (λ_{n}^{f_{\infty}} (x) - ε)),

which means that

y \in B (x, δ \cdot exp (- n \cdot (λ_{n}^{f_{\infty}} (x) - ε)))

In the second step, we are going to prove the inclusion

B (x, δ \cdot e^{- n (λ_{n}^{f_{\infty}} (x) + ε)}) \subset B (x, n, δ)

. Since the dynamical solenoid is expanding,

log a_{i} > 0

, for any

i \geq 1

. In particular, it follows that for any

1 \leq i \leq j \leq n

we have

(n - i + 1) λ_{i, n}^{f_{\infty}} \geq (n - j + 1) λ_{j, n}^{f_{\infty}}

. Setting

i = 1

, we obtain

δ e^{- n (λ_{n}^{f_{\infty}} + ε)} \leq δ e^{- (n - j + 1) (λ_{j, n}^{f_{\infty}} + ε)} < δ, for j = 1, \dots, n .

(16)

We fix

x \in X

and consider a point

y \in B (x, δ e^{- n (λ_{n}^{f_{\infty}} (x) + ε)})

. Applying the identities

x_{n} = x

y_{n} = y

and definition of

B (x, n, δ)

, we have

d (x_{n}, y_{n}) = d (x, y) < δ e^{- n (λ_{n}^{f_{\infty}} (x) + ε)} .

Due to (16), for any

j = 1, \dots, n

, we obtain the inequality

d (x_{n}, y_{n}) < δ e^{- (n - j + 1) (λ_{j, n}^{f_{\infty}} + ε)} .

From the first inequality of Corollary 7 (b), we successively obtain the following.

For

n \geq 1

\begin{matrix} d (x_{n - 1}, y_{n - 1}) & = & d (f_{n} (x_{n}), f_{n} (y_{n})) \\ < & d (x_{n}, y_{n}) \cdot e^{λ_{n, n}^{f_{\infty}} (x_{n}) + ε} \\ = & d (x_{n}, y_{n}) e^{λ_{n, n}^{f_{\infty}} (x_{n}) + ε} < δ . \end{matrix}

For

n \geq 2

\begin{matrix} d (x_{n - 2}, y_{n - 2}) & = & d (f_{n - 1} (x_{n - 1}), f_{n - 1} (y_{n - 1})) \\ < & d (x_{n - 1}, y_{n - 1}) \cdot e^{a_{n - 1} (x_{n - 1}) + ε} \\ = & d (x_{n}, y_{n}) \cdot e^{a_{n} (x_{n}) + ε} \cdot e^{a_{n - 1} (x_{n - 1}) + ε} \\ = & d (x_{n}, y_{n}) \cdot e^{2 (λ_{n - 1, n}^{f_{\infty}} (x_{n}) + ε)} < δ . \end{matrix}

For

n \geq j

\begin{matrix} d (x_{n - j}, y_{n - j}) & = d (f_{n - j + 1} (x_{n - j + 1}), f_{n - j + 1} (y_{n - j + 1})) \\ < d (x_{n - j + 1}, y_{n - j + 1}) \cdot e^{a_{n - j + 1} (x_{n - j + 1}) + ε} \\ = d (x_{n}, y_{n}) \cdot e^{a_{n} (x_{n}) + ε} \dots e^{a_{n - j + 1} (x_{n - j + 1}) + ε} \\ = d (x_{n}, y_{n}) \cdot e^{j (λ_{n - j + 1, n}^{f_{\infty}} (x_{n}) + ε)} < δ . \end{matrix}

We have shown that for any

k = 0, \dots, n

the inequality

d (x_{k}, y_{k}) < δ

holds, which is equivalent to

y \in B (x, n, δ)

. □

6. Proof of Theorem 4

Let

t^{*} > 0

denote the unique root of the pressure function

t \mapsto P_{Z} (- t \cdot log a_{\infty})

, which exists and is positive by Theorem 3. Hence, by Corollary 6 (c), we have

\begin{matrix} P_{Z} (- t \cdot log a_{\infty}) & < & 0, for t > t^{*} \end{matrix}

(17)

\begin{matrix} P_{Z} (- t \cdot log a_{\infty}) & > & 0, for t < t^{*} \end{matrix}

(18)

First, we will show that

{dim}_{H} (Z) \leq t^{*}

. For a natural number

m \in N

and

α > 0

, let us define a set

Z_{m} = \{x \in Z : λ_{n}^{f_{\infty}} (x) > α, for any n \geq m\} .

Since

(X_{\infty}, f_{\infty}, a_{\infty})

is a DCSE-solenoid, there exists

β > 0

such that

\bar{λ^{f_{\infty}}} < β

. Note (cf. Section 3) that

A ((α, β)) = {x \in X : α < \underset{̲}{λ^{f_{\infty}}} (x) \leq \bar{λ^{f_{\infty}}} (x) < β} .

Directly from the definition of

Z_{m}

and the inclusion

Z \subset A ((α, β))

, we obtain

Z_{m} \subset Z_{m + 1}, ⋃_{m \in N} Z_{m} = Z .

(19)

We fix

t > t^{*} \geq 0

. Due to (17), we can choose

0 < ε < α

, such that

\begin{matrix} - t \cdot ε > P_{Z} (- t \cdot log a_{\infty}) \geq P_{Z_{m}} (- t \cdot log a_{\infty}), \end{matrix}

(20)

where the second inequality is a consequence of Lemma 9. By Lemma 12, there exists

δ^{'} > 0

such that for any

x \in Z_{m}

n \geq m

, and any

0 < δ < δ^{'}

\begin{matrix} diam B (x, n, δ) & \leq 2 δ exp (- n (λ_{n}^{f_{\infty}} (x) - ε)) \\ < 2 δ exp (- n (α - ε)) . \end{matrix}

(21)

This implies that for any

N > m

and any

0 < δ < δ^{'}

P (Z_{m}, N, δ) \subset D (Z_{m}, 2 δ exp (- N (α - ε))) .

(22)

Note that the first inequality in (21) leads to the estimation

{[\frac{1}{2 δ} diam B (x, n, δ)]}^{t} \leq exp (- n t (log λ_{n}^{f_{\infty}} (x) - ε)), n \geq m, x \in Z_{m} .

Therefore, for

N > m

and any

I \in I (Z_{m}, N, δ)

applying definitions presented in Section 3, we have

\begin{matrix} Φ (- t ε, - t log a_{\infty}) \\ = \sum_{(x_{i}, n_{i}) \in I} exp (- n_{i} t (log λ_{n_{i}}^{f_{\infty}} (x_{i}) - ε)) \\ \geq \sum_{(x_{i}, n_{i}) \in I} {[\frac{1}{2 δ} diam B (x_{i}, n_{i}, δ)]}^{t} \\ \geq inf \{\frac{1}{{(2 δ)}^{t}} \sum_{V \in V} {(diam V)}^{t} : V \in D (Z_{m}, 2 δ exp (- N (α - ε))\} \\ = \frac{1}{{(2 δ)}^{t}} \cdot H^{t} (Z_{m}, 2 δ exp (- N (α - ε))) \end{matrix}

This implies that

\begin{matrix} m_{P} (Z_{m}, - t ε, - t log a_{\infty}, N, δ) \\ = inf {Φ (- t ε, - t log a_{\infty}, N, δ) : I \in I (Z_{m}, N, δ)} \\ \geq \frac{1}{{(2 δ)}^{t}} \cdot H^{t} (Z_{m}, 2 δ exp (- N (α - ε))) . \end{matrix}

Due to the inequality

α - ε > 0

, in the limit with

N \to \infty

yields

\begin{matrix} m_{P} (Z_{m}, - t ε, - t log a_{\infty}, δ) \\ = lim_{N \to \infty} m_{P} (Z_{m}, - t ε, - t log a_{\infty}, N, δ) \\ \geq \frac{1}{{(2 δ)}^{t}} lim_{N \to \infty} H^{t} (Z_{m}, 2 δ exp (- N (α - ε))) \\ = \frac{1}{{(2 δ)}^{t}} \cdot H^{t} (Z_{m}) . \end{matrix}

We claim that

\begin{matrix} H^{t} (Z_{m}) = 0 . \end{matrix}

(23)

Otherwise, if

H^{t} (Z_{m}) > 0

then by Lemma 8,

m_{P} (Z_{m}, s, - t log a_{\infty}, δ) = + \infty

for all real

s < - t ε

. This means that

P_{Z_{m}} (- t log a_{\infty}, δ) = sup {s \in R : m_{P} (Z_{m}, s, - t log a_{\infty}, δ) = + \infty} \geq - t ε .

Then, by (8), we have

P_{Z_{m}} (- t log a_{\infty}) = sup {P_{Z_{m}} (- t log a_{\infty}, δ) : δ > 0} \geq - t ε,

which contradicts the inequality (20). Thus, we have proven (23).

By properties of

t -

Hausdorff measure, we have

H^{t} (Z) = H^{t} (⋃ Z_{m}) = sup {H^{t} (Z_{m}) : m \geq 1} = 0,

which means that

{dim}_{H} (Z) \leq t

for any

t > t^{*}

. Thus, we obtain the inequality

{dim}_{H} (Z) \leq t^{*}

Let us proceed to prove the inequality

{dim}_{H} (Z) \geq t^{*}

. This inequality holds trivially when

t^{*} = 0

. Therefore, assume that

t^{*} > 0

. Take

0 < t < t^{*}

. Then, due to (18), there exists

ε > 0

such that

0 < t ε < P_{Z} (- t log a_{\infty}) .

Hence, by Lemma 5 and the definition of topological pressure, there is

\bar{δ} > 0

such that for any

0 < δ < \bar{δ}

0 < t ε < P_{Z} (- t log a_{\infty}, δ) .

(24)

Let

b = sup {a_{n} (x) : x \in X, n \geq 1}

. Since

(X_{\infty}, f_{\infty}, a_{\infty})

is a DCSE-solenoid,

1 < b < \infty

. Notice that for any

1 \leq i \leq j

and any

x \in X

λ_{i, j}^{f_{\infty}} (x) = \frac{1}{j - i + 1} \sum_{k = i}^{j} log a_{k} (f_{k, j} (x)) \leq \frac{1}{j - i + 1} \sum_{k = i}^{j} log b = log b .

For

ε > 0

chosen above, we fix

δ^{'} > 0

given by Lemma 12. Let

0 < δ < δ^{'}

. For

1 \leq i \leq j

, we set

s_{i, j} (x) = δ exp (- (j - i + 1) (λ_{i, j}^{f_{\infty}} (x) + ε)) .

In particular,

s_{1, n} (x) = δ exp (- n (λ_{1, n}^{f_{\infty}} + ε)) = δ exp (- n (λ_{n}^{f_{\infty}} (x) + ε)) .

Since, for any natural

n \geq 1

log a_{n} > 0

, we obtain the estimation

s_{i, n} (x) < δ exp (- (n - i + 1) ε)), for n \geq i .

Hence, it follows that for any

x \in X

lim_{n \to \infty} s_{i, n} (x) = 0 .

(25)

Moreover, this convergence is uniform. We have

\begin{matrix} \frac{s_{2, n} (x)}{s_{1, n} (x)} & = \frac{δ exp (- (n - 1) (λ_{2, n}^{f_{\infty}} (x) + ε))}{δ exp (- n (λ_{1, n}^{f_{\infty}} (x) + ε))} \\ = \frac{exp (- S_{2, n}^{f_{\infty}} (log a_{\infty}) (x) - (n - 1) ε)}{exp (- S_{1, n}^{f_{\infty}} (log a_{\infty}) (x) - n ε)} \\ = exp [(- \sum_{k = 2}^{n} log a_{k} (f_{k, n} (x)) - (n - 1) ε) \\ - (- \sum_{k = 1}^{n} log a_{k} (f_{k, n} (x)) - n ε)] \\ = exp (log a_{1} (f_{1, n} (x)) + ε) . \end{matrix}

By the definition of b, we conclude that

log b \geq log (a_{1} (f_{1, n} (x))

. Therefore,

1 < exp (log a_{1} (f_{1, n} (x)) + ε) \leq exp (log b + ε) .

Consequently we obtain

e^{- (log b + ε)} s_{1, n} (x) < e^{- (log b + ε)} s_{2, n} (x) \leq s_{1, n} (x) < s_{2, n} (x) .

Hence, by (25), there exists

\bar{r} > 0

such that for any

x \in Z

and any

0 < r < \bar{r}

, there exists

n \geq 2

, such that

exp (- (log b + ε)) s_{1, n} (x) \leq r < s_{1, n} (x) = δ exp (- n (λ_{n}^{f_{\infty}} (x) + ε)) .

Thus,

\begin{matrix} r & \geq e^{- (log b + ε)} s_{1, n} (x) \\ = e^{- (log b + ε)} δ exp (- n (λ_{n}^{f_{\infty}} (x) + ε)) \\ \geq e^{- (log b + ε)} δ exp (- n (log b + ε)) \\ = δ exp (- (n + 1) (log b + ε)), \end{matrix}

or, equivalently (after taking the logarithms of both sides),

n \geq N (r, δ) and n \in N,

where

N (r, δ) = \frac{- log r + log δ}{log b + ε} - 1 .

Note that the function

r \mapsto N (r, δ)

is strictly decreasing with respect to

r > 0

and

{lim}_{r \to 0} N (r, δ) = + \infty

Let

0 < r < \bar{r}

. Suppose that

{B (x_{i}, r_{i}) : (x_{i}, r_{i}) \in I}

is at most countable cover of the set Z with balls of positive radii

r_{i} < r

and centers

x_{i} \in Z

. Then, due to previous considerations, there exists a sequence of natural numbers

(n_{i})

such that

\begin{matrix} δ exp (- (log b + ε)) s_{1, n_{i}} (x_{i}) \leq & r_{i} & < s_{1, n_{i}} (x_{i}), \\ N (r, δ) < N (r_{i}, δ) \leq & n_{i}, \end{matrix}

where

s_{1, n_{i}} (x_{i}) = δ exp (- n_{i} (λ_{n_{i}}^{f_{\infty}} (x_{i}) + ε)) .

Hence, by Lemma 12, we obtain the inclusion

B (x_{i}, r_{i}) \subset B (x_{i}, n_{i}, δ)

. This means that the cover

{B (x_{i}, r_{i})}

is a refinement of the cover

{B (x_{i}, n_{i}, δ)}

of the set Z by dynamical balls. Since

{B (x_{i}, n_{i}, δ)} \in P (Z, N (r, δ), δ)

, we proceed to the following estimations:

\begin{matrix} M_{H} (Z, t, r) = inf \{\sum_{(x_{i}, r_{i}) \in I} {(2 r_{i})}^{t} : Z \subset ⋃_{(x_{i}, r_{i}) \in I} B (x_{i}, r_{i}), r_{i} < r, x_{i} \in Z\} \\ \geq inf \{\sum_{(x_{i}, r_{i}) \in I} 2^{t} e^{- (log b + ε) t} s_{1, n_{i}}^{t} (x_{i}) : Z \subset ⋃_{(x_{i}, r_{i}) \in I} B (x_{i}, r_{i}), r_{i} < r, x_{i} \in Z\} \\ = inf \{\sum_{(x_{i}, r_{i}) \in I} {(2 δ)}^{t} e^{- (log b + ε) t} e^{- t n_{i} (λ_{n_{i}}^{f_{\infty}} (x_{i}) + ε)} : Z \subset ⋃_{(x_{i}, r_{i}) \in I} B (x_{i}, r_{i}), r_{i} < r, x_{i} \in Z\} \\ = {(2 δ)}^{t} e^{- (log b + ε) t} inf \{\sum_{(x_{i}, r_{i}) \in I} e^{- t n_{i} (λ_{n_{i}}^{f_{\infty}} (x_{i}) + ε)} : Z \subset ⋃_{(x_{i}, r_{i}) \in I} B (x_{i}, r_{i}), r_{i} < r, x_{i} \in Z\} \\ \geq {(2 δ)}^{t} e^{- (log b + ε) t} inf \{Φ (t ε, I, - t log a_{\infty}) : I \in I (Z, N (r, δ), δ)\} \\ = {(2 δ)}^{t} e^{- (log b + ε) t} m_{P} (Z, t ε, - t log a_{\infty}, N (r, δ), δ) . \end{matrix}

Therefore, we obtain

\begin{matrix} M_{H} (Z, t) & = lim_{r \to 0} M_{H} (Z, t, r) \\ \geq {(2 δ)}^{t} e^{- (log b + ε) t} lim_{r \to 0} m_{P} (Z, t ε, - t log a_{\infty}, N (r, δ), δ) \\ = {(2 δ)}^{t} e^{- (log b + ε) t} m_{P} (Z, t ε, - t log a_{\infty}, δ) = + \infty, \end{matrix}

where the last equality follows from (5) and (7). Therefore,

{dim}_{H} (Z) > t

. Since t was an arbitrary number from the interval

(0, t^{*})

, we conclude that

{dim}_{H} (Z) \geq t^{*}

. This completes the proof of the theorem.

Corollary 1 in Section 1 is a direct consequence of Theorem 4.

7. Apendix: Conformality vs. Strong Conformality

The following example compares dilatation with strong dilatation. On a plane, we consider four points: A(−2,0), B(2,0), O(0,0), Q(0,−2). Let

I_{1}

be a segment AB, while

I_{2}

a segment OQ. Consider a metric space

T = I_{1} \cup I_{2}

equipped with the city metric

d .

| \bar{P R} |

, we mean the length of the segment PR. In particular, we have

(1): if $x \in \bar{A B}$ , then $d (x, Q) = | \bar{x O} | + | \bar{O Q} | = | \bar{x O} | + 2$ ;
(2): if $y \in \bar{O Q}$ , then $d (y, Q) = | \bar{y Q} |$ .

Let

f : T \to T

be a homothety with ratio

1 / 2

and fixed point Q; it means that for any

x \in T

, there exists a unique point

f (x) \in \bar{O Q}

with

d (f (x), Q) = \frac{1}{2} d (x, Q)

. Now we calculate a dilatation

a_{w} (x)

for any

x \in T .

Notice that for

x, y \in T

and f, we have

d (f (x), Q) = \frac{1}{2} d (x, Q), d (f (y), Q) = \frac{1}{2} d (y, Q) .

We fix

x \in T

. By the definition of

(T, d)

, it easily follows that there exists a neighborhood

Γ

of x such that for any

y \in Γ

\begin{matrix} d (f (x), f (y)) & = | d (f (x), Q) - d (f (y), Q) | \\ = \frac{1}{2} | d (x, Q) - d (y, Q) | = \frac{1}{2} d (x, y) . \end{matrix}

Therefore, for any

x \in T

, we obtain

a_{w} (x) = lim_{y \to x} \frac{d (f (x), f (y))}{d (x, y)} = \frac{1}{2},

f : T \to T

is a conformal map. Consider a function of two variables

ζ : T \times T \to R

defined by the formula

ζ (x, y) = \{\begin{matrix} log d (f (x), f (y)) - log d (x, y), & if x \neq y \\ log a_{w} (x), & if x = y . \end{matrix}

We fix

x_{0} = (0, 0) \in T

. By definition, the function

ζ (x, y)

is continuous at

(x_{0}, x_{0})

if and only if for any sequences

x_{n} \to x_{0}

and

y_{n} \to x_{0}

the limit

lim_{\begin{matrix} x_{n} \to x_{0} \\ y_{n} \to x_{0} \end{matrix}} ζ (x_{n}, y_{n}) = ζ (x_{0}, x_{0}) .

In particular, for the sequences

x_{n} = (- \frac{2}{n}, 0)

and

y_{n} = (\frac{4}{n}, 0)

we obtain

d (x_{n}, y_{n}) = \frac{6}{n}

. Moreover,

f (x_{n}) = (0, \frac{1}{n} - 1)

and

f (y_{n}) = (0, \frac{2}{n} - 1)

. Thus,

d (f (x_{n}), f (y_{n})) = \frac{1}{n}

. Consequently,

lim_{\begin{matrix} x_{n} \to x_{0} \\ y_{n} \to x_{0} \end{matrix}} ζ (x_{n}, y_{n}) = - log 6 .

However,

ζ (x_{0}, x_{0}) = log a_{w} (x_{0}) = - log 2,

which yields discontinuity of

ζ

at the point

(x_{0}, x_{0})

. Moreover, the strong dilatation of the function

f : T \to T

is not even defined at

x_{0} .

It follows that

f : T \to T

is an example of a conformal map which is not a strong conformal one.

Finally, we note that the strong conformality of f implies that

A_{f} : X \times X \to R

is well defined and uniformly continuous on some neighborhood of the diagonal (see Lemma 11). This fact is a key step in the proof of crucial Lemma 12.

Author Contributions

Conceptualization, A.B. and W.K.; Methodology, A.B. and W.K.; Investigation, A.B., W.K. and A.M. All authors have read and agreed to the published version of the manuscript.

Funding

The first author was partially supported by the inner Lodz University Grant 11/IDUB/DOS/2021.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Acknowledgments

It is a pleasure to thank Paweł Walczak and Maciej Czarnecki for the useful discussions and suggestions on the topic of the paper. We are also very grateful to the anonymous referees for their careful reading of the manuscript and for their comments that helped us to improve the paper.

Conflicts of Interest

The authors declare no conflicts of interest.

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Biś, A.; Kozłowski, W.; Marczuk, A. Bowen’s Formula for a Dynamical Solenoid. Entropy 2024, 26, 979. https://doi.org/10.3390/e26110979

AMA Style

Biś A, Kozłowski W, Marczuk A. Bowen’s Formula for a Dynamical Solenoid. Entropy. 2024; 26(11):979. https://doi.org/10.3390/e26110979

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Biś, Andrzej, Wojciech Kozłowski, and Agnieszka Marczuk. 2024. "Bowen’s Formula for a Dynamical Solenoid" Entropy 26, no. 11: 979. https://doi.org/10.3390/e26110979

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Bowen’s Formula for a Dynamical Solenoid

Abstract

1. Introduction and Statement of Results

2. Conformal Mappings

3. Dynamical Solenoids

3.1. Lyapunov Exponents for Dynamical Solenoids

3.2. Topological Pressure for Dynamical Solenoids

3.3. Topological Entropy and Pressure for Dynamical Solenoids

3.4. Proof of Theorem 3

4. Hausdorff Dimension

5. Properties of Dynamical Balls of DCSE-Solenoids

6. Proof of Theorem 4

7. Apendix: Conformality vs. Strong Conformality

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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