Let be a weakly conformal iterated function system on ℝ^d with attractor K. Let π be the canonical projection. In this paper we define a new concept called “projection pressure” P_π(φ) for φ ∈ C(Σ) and show the variational principle about the projection pressure under AWSC. Furthermore, we check that the zero of “projection pressure” still satisfies Bowen′s equation. Using the root of Bowen′s equation, we can get the Hausdorff dimension of the attractor K.

1. Introduction

Let be a family of contractive maps on a nonempty closed set X ⊂ ℝ^d. Following Barnsley [1], we say that is an iterated function system (IFS) on X. Hutchinson [2] showed that there is a unique nonempty compact set K ⊂ X, called the attractor of , such that .

There are many references to compute the Hausdorff dimension of K or the Hausdorff dimension of multifractal spectrum, such as, [3–5]. Thermodynamic formalism played a significant role when we try to compute the Hausdorff dimension of K, especially the Bowen’s equation. Usually, we call P_J(tψ) = 0 the Bowen’s equation, where P_J is the topological pressure of the map f : J → J, and ψ is the geometric potential ψ(z) = log | f′(z)|. The root of Bowen’s equation always approaches the Hausdorff dimension of some sets. In [6], Bowen first discovered this equation while studying the Hausdorff dimension of quasicircles. Later Ruelle [7], Gatzouras and Peres [8] showed that Bowen’s equation gives the Hausdorff dimension of J whenever f is a C¹ conformal map on a Riemannian manifold and J is a repeller. According to the method for calculating Hausdorff dimension of cookie-cutters presented by Bedford [9], Keller discussed the relation between classical pressure and dimension for IFS [10]. He concluded that if is a conformal IFS satisfying the disjointness condition with local energy function ψ, then the pressure function has a unique zero root t₀ = dim _HK. In 2000, using the definition of Carathe’ odory dimension characteristics, Barreira and Schmeling [11] introduced the notion of the u-dimension dim _uZ for positive functions u, showing that dim _uZ is the unique number t such that P_Z(−tu) = 0.

On the progress of calculating dim _HK, [3–5] depend on the open set condition and separable condition. In fact, there are a lot of examples that do not satisfy this disjointness condition. Rao and Wen once discussed a kind of self-similar fractal with overlap structure called λ-Cantor set [12].

In order to study the Hausdorff dimension of an invariant measure μ for conformal and affine IFS with overlaps, Feng and Hu introduce a notion projection entropy (see [13]), which plays the similar role as the classical entropy of IFS satisfying the open set condition, and it becomes the classical entropy if the projection is finite to one.

Bedford pointed out that the Bowen’s equation works if three elements are present: (i) conformal contractions, (ii) open set conditions, and (iii) subshift of finite-type (Markov) structure. Chen and Xiong [14] proved that subshift of finite-type (Markov) structure can be replaced by any subshift structure. In [15, 16], the authors defined projection pressure for two different types of IFS. In this paper, we consider projection pressure under asymptotically weak separation condition (AWSC) and check that Bowen’s equation still holds.

2. The Projection Pressure for AWSC: Definition and Variational Principle

Let

be an IFS on a closed set X ⊂ ℝ^d. Denote by K its attractor. Let Σ = {1, …, l} ^ℕ associated with the left shift σ. Let M_σ(Σ) denote the space of σ-invariant measure on Σ, endowed with the weak-star topology, C(X) the space of real-valued continuous functions of X, and π : Σ → K be the canonical projection defined by

(2.1)

A measure μ on K is called invariant (resp., ergodic) for the IFS if there is and invariant (resp., ergodic) measure ν on Σ such that μ = ν∘π⁻¹.

Let (Ω, ℱ, ν) be a probability space. For a sub-σ-algebra 𝒜 of ℱ and f ∈ L¹(Ω, ℱ, ν), we denote by E_ν(f | 𝒜) the conditional expectation of f given 𝒜. For countable ℱ-measurable partition ξ of Ω. We denote by I_ν(ξ | 𝒜) the conditional information of ξ given 𝒜, which is given by the formular:

(2.2)

where 𝒳_A denote the characteristic function on 𝒜.

The conditional entropy of ξ given 𝒜, written H_ν(ξ | 𝒜) is defined by the formula H_ν(ξ | 𝒜) = ∫I_ν(ξ | 𝒜)dν.

The above information and entropy are unconditional when 𝒜 = 𝒩, the trivial σ-algebra consisting of sets of measure zero and one, and in this case we write

(2.3)

Now we consider the space

, where

is the Borel σ-algebra on Σ and m ∈ M_σ(Σ). Let 𝒫 denote the Borel partition:

(2.4)

of Σ, where

. Let ℐ denotes the σ-algebra:

(2.5)

For convenience, we use γ to denote the Borel σ-algebra ℬ(ℝ^d) of ℝ^d. For f ∈ C(X), denote ∥f∥ = sup _x∈Xf(x) and , for all x ∈ X. Let Σ_n = {[b]:[b] = (x₁, x₂, …, x_n), x_i ∈ Σ, i = 1, …, n}.

Definition 2.1. For any m ∈ M_σ(Σ), we call

(2.6)

the projection entropy of m under π w.r.t {S_i}, and we call

(2.7)

the local projection entropy of m at x under π w.r.t

, where f denote the function I_m(𝒫 | σ⁻¹π⁻¹γ) − I_m(𝒫 | σ⁻¹γ).

It is clear that h_π(σ, m) = ∫h_π(σ, m, x)dm(x).

The following Lemma 2.2 gives the relation between the projection entropy and the classical entropy and the basic properties of the new entropy which are similar to the classical entropy’s. For more details we can see Theorem 2.2 in [13].

Lemma 2.2. Let be an IFS. Then

(i)
For any m ∈ M_σ(Σ), one has 0 ≤ h_π(σ, m) ≤ h(σ, m), where h(σ, m) denotes the classical measure-theoretic entropy of m associated with σ.
(ii)
The map m ↦ h_π(σ, m) is affine on M_σ(Σ). Furthermore if m = ∫νdℙ(ν) is the ergodic decomposition of m, one has
(2.8)
(iii)
For any m ∈ M_σ(Σ), one has
(2.9)
for m-a.e. x ∈ Σ, where h(σ, m, x) denotes the local entropy of m at x, that is, h(σ, m, x) = I_m(𝒫∣σ⁻¹ℬ(Σ))(x).

Definition 2.3. Let k ∈ ℕ and . Define

(2.10)

The term h_π(σ^k, ν) can be viewed as the projection measure-theoretic entropy of ν w.r.t. the IFS . The following lemma exploits the connection between h_π(σ^k, ν) and h_π(σ, m), where .

Lemma 2.4. Let k ∈ ℕ and . Set . Then m is σ-invariant, and h_π(σ, ν) = (1/k)h_π(σ^k, ν) = (1/k)h_π(σ^k, m).

Proof. See Proposition 4.3 in [13].

Definition 2.5. An IFS on a compact set X ⊂ ℝ^d is said to satisfy the asymptotically weak separation condition (AWSC), if

(2.11)

where t_n is given by

(2.12)

here K is the attractor of

Lemma 2.6. Let be an IFS with attractor K. Suppose that Ω is a subset of {1, …, l} such that there is a map g: {1, …, l} → Ω so that

(2.13)

Let

denote the one-side full shift over Ω. Define G: Σ → Ω^ℕ by

. Then

(i)
K is also the attractor of {S_i} _i∈Ω. Moreover, if one lets denote the canonical projection w.r.t. {S_i} _i∈Ω, then one has .
(ii)
Let m ∈ M_σ(Σ). Then . Furthermore, .

Proof. See Lemma 4.23 in [13].

Lemma 2.7. Let be an IFS with attractorK ⊂ ℝ^d. Assume that

(2.14)

for some k ∈ ℕ and each x ∈ ℝ^d. Then h_π(σ, m) ≥ h(σ, m) − log k for any m ∈ M_σ(Σ).

Proof. See Lemma 4.21 in [13].

Lemma 2.8. Let a₁, a₂, …, a_k be given real numbers. If p_i ≥ 0 and , then and equality holds iff .

Proof. See Lemma 9.9 in [17].

For convenience, for n ∈ ℕ, write Σ_n = {1, …, l} ⁿ. According to Lemma 2.6 there is a set Ω_n ⊂ Σ_n and a map g : Σ_n → Ω_n such that S_u = S_g(u) for u ∈ Σ_n. Let

denote the one-sided full shift space over the alphabet

and ξ_n denote the natural generator. Let

be defined by

(2.15)

Theorem 2.9. Suppose an IFS satisfies the AWSC with attractor K and f : Σ → ℝ is continuous. Then

(2.16)

Proof. We divided the proof into two steps.

Step 1.

(2.17)

For arbitrary n ∈ ℕ, m ∈ M_σ(Σ), then

. By Lemma 2.8, we have

(2.18)

By Lemma 2.2(i) and Lemma 2.6(ii), divided by n yields

(2.19)

By the arbitrariness of m and n, we have Step 1.

Step 2.

(2.20)

By the continuity of f, for arbitrary ϵ > 0, there exists N ∈ ℕ such that for arbitrary a_N ∈ Σ_N and any x, y ∈ a_N, we have

(2.21)

Now, for any n ∈ ℕ and a_n+N ∈ Σ_n+N

(2.22)

Define a Bernoulli measure ν on by

(2.23)

Then ν can be viewed as a σ^n+N-invariant measure on Σ (by viewing

as a subset of Σ). By Lemma 2.6, we have

. Define

. We have

(2.24)

Let k = n + N and let n → ∞, then k → ∞. We have

(2.25)

Since ϵ is arbitrary, we finish the proof of Step 2.

Definition 2.10. If an IFS satisfies AWSC with attractor K and f ∈ C(Σ). We call

(2.26)

the projection pressure of f under π w.r.t.

It is clearly that, if f = 0, we have the same result of Lemma 9.1 in [13].

Corollary 2.11. lim _n→∞ (log #{S_u : u ∈ Σ_n}/n) = sup {h_π(σ, m) : m ∈ M_σ(Σ)}.

3. Application for Projection Pressure

Definition 3.1. is called a C¹ IFS on a compact set X ⊂ ℝ^d if each S_i extends to a contracting C¹-diffeomorphism S_i : U → S_i(U) ⊂ U on an open set U⊃X.

For any d × d real matrix M, we use ∥M∥ to denote the usual norm of M and ⫾M⫾ the smallest singular value of M, that is,

(3.1)

Definition 3.2. The IFS is conformal if for every i ∈ {1,2, …, l}, (1) S_i : U → S_i(U) is C¹, (2) for all x ∈ U, and (3) for all x ∈ U, y ∈ ℝ^d.

Definition 3.3. Let be a C¹ IFS. For , the upper and lower Lyapunov exponents of at x are defined, respectively, by

(3.2)

where

denote the differential of

at πσⁿx. When

, the common value, denoted as λ(x), is called the Lyapunov exponents of

at x.

It is easy to check that both and are positive-valued σ-invariant functions on Σ (i.e., and ).

Definition 3.4. A C¹ IFS is said to be weakly conformal if

(3.3)

converges to 0 uniformly on Σ as n tends to ∞.

If IFS is weakly conformal, by Birkhoff’s ergodic theorem, we can conclude .

Lemma 3.5. Let K be the attractor of a weakly conformal IFS . Then we have

(3.4)

(3.5)

(3.6)

(3.7)

Proof. See Theorem 2.13 in [13].

Theorem 3.6. Let be a weakly conformal IFS satsifying AWSC. Let and π : Σ → K be the canonical projection. Then dim _HK is the unique root of P_π(tψ) = 0.

Proof. According to Theorem 2.9, we have

(3.8)

Let t₀ = sup {h_π(σ, m)/∫λ dm : m ∈ M_σ(Σ)}, according to (3.7) t₀ = dim _HK.

First we show P_π(tψ) is decreased with respect to t. If 0 ≤ t₁ ≤ t₂, then for any m ∈ M_σ(Σ), we have h_π(σ, m)+∫t₁ψ dm ≥ h_π(σ, m)+∫t₂ψ dm. Hence according to variational principle, we have P_π(t₁ψ) ≥ P_π(t₂ψ).

As t₀ ≥ h_π(σ, m)/∫λ dm for all m ∈ M_σ(Σ), h_π(σ, m) + t₀∫ψ dm ≤ 0 for all m ∈ M_σ(Σ), whence P_π(t₀ψ) ≤ 0.

However, P_π(0) > 0, the existence of a positive zero t₁ for t ↦ P_π(tψ) follows from the intermediate value theorem, that is, sup {h_π(σ, m)+∫t₁ψ dm, m ∈ M_σ(Σ)} = 0.

For all ϵ > 0 there is a m ∈ M_σ(Σ) such that h_π(σ, m)+∫t₁ψ dm ≥ −ϵ. Thus, t₁ ≤ h_π(σ, m)/∫−ψ dm + ϵ/∫−ψ dm ≤ sup {h_π(σ, m)/∫−ψ dm, m ∈ M_σ(Σ)} + ϵ/∫−ψ dm, let ϵ → 0 we have t₁ ≤ t₀. And t₀ ≤ t₁ as P_π(t₁ψ) = 0 implies h_π(σ, m) + t₁∫ψ dμ ≤ 0 for all μ ∈ M_σ(Σ). So any root of P_π(tψ) = 0 is equal to dim _HK.

Acknowledgments

The author Acknowledges the support by the National Natural Science Foundation of China (10971100) and National Basic Research Program of China (973 Program) (2007CB814800).

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The Projection Pressure for Asymptotically Weak Separation Condition and Bowen′s Equation

Abstract

1. Introduction

2. The Projection Pressure for AWSC: Definition and Variational Principle

3. Application for Projection Pressure

Acknowledgments

References

References

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The Projection Pressure for Asymptotically Weak Separation Condition and Bowen′s Equation

Abstract

1. Introduction

2. The Projection Pressure for AWSC: Definition and Variational Principle

3. Application for Projection Pressure

Acknowledgments

References

References

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