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

Let {Si}i 1 be a weakly conformal iterated function system on R with attractor K. Let π be the canonical projection. In this paper we define a new concept called “projection pressure” Pπ 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.


Introduction
Let {S i : X → X} l i 1 be a family of contractive maps on a nonempty closed set X ⊂ R d .Following Barnsley 1 , we say that {S i } l i 1 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 {S i } l i 1 , such that K ∪ l i 1 S i K .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 C 1 conformal map on a Riemannian manifold and J is a repeller.According to the method for calculating Hausdorff dimension of cookiecutters presented by Bedford 9 , Keller discussed the relation between classical pressure and dimension for IFS 10 .He concluded that if {S i } l i 1 is a conformal IFS satisfying the disjointness condition with local energy function ψ, then the pressure function has a unique zero root t 0 dim H K. In 2000, using the definition of Carathe' odory dimension characteristics, Barreira and Schmeling 11 introduced the notion of the u-dimension dim u Z for positive functions u, showing that dim u Z is the unique number t such that P Z −tu 0. On the progress of calculating dim H K, 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.

The Projection Pressure for AWSC: Definition and Variational Principle
Let {S i } l i 1 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 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 ∈ L 1 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: 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 σalgebra consisting of sets of measure zero and one, and in this case we write Now we consider the space Σ, B Σ , m , where B Σ is the Borel σ-algebra on Σ and m ∈ M σ Σ .Let P denote the Borel partition: Let I denotes the σ-algebra: For convenience, we use γ to denote the Borel the projection entropy of m under π w.r.t {S i }, and we call the local projection entropy of m at x under π w.r.t {S i } l i 1 , where f denote the function

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 {S i } l i 1 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 iii For any m ∈ M σ Σ , one has 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

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 m Proof.See Proposition 4.3 in 13 .
Definition 2.5.An IFS {S i } l i 1 on a compact set X ⊂ R d is said to satisfy the asymptotically weak separation condition AWSC , if where t n is given by here K is the attractor of {S i } l i 1 .
Lemma 2.6.Let {S i } l i 1 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 S i S g i i 1, . . ., l .

2.13
Let Ω N , σ denote the one-side full shift over Ω. Define G: i 0 e a i and equality holds iff p i e a i / k j 1 e a j .
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 Ω N n , T denote the one-sided full shift space over the alphabet Ω N n and ξ n denote the natural generator.Let G : Σ → Ω N n be defined by Theorem 2.9.Suppose an IFS {S i } l i 1 satisfies the AWSC with attractor K and f : Proof.We divided the proof into two steps.

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.
By the continuity of f, for arbitrary > 0, there exists N ∈ N such that for arbitrary a N ∈ Σ N and any x, y ∈ a N , we have Now, for any n ∈ N and

2.22
Define a Bernoulli measure ν on Ω N n N by

2.23
Then ν can be viewed as a σ n N -invariant measure on Σ by viewing Ω N n as a subset of Σ .By Lemma 2.6, we have

2.24
Let k n N and let n → ∞, then k → ∞.We have Since is arbitrary, we finish the proof of Step 2.
Definition 2.10.If an IFS {S i } l i 1 satisfies AWSC with attractor K and f ∈ C Σ .We call 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 .

Application for Projection Pressure
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, If IFS {S i } l i 1 is weakly conformal, by Birkhoff's ergodic theorem, we can conclude λ x dm − log S x 1 πσx dm − log S x 1 πσx dm.

Lemma 3.5. Let K be the attractor of a weakly conformal IFS {S
Proof.See Theorem 2.13 in 13 .
Theorem 3.6.Let {S i x } l i 1 be a weakly conformal IFS satsifying AWSC.Let ψ x log S x 1 πσ x : Σ → R and π : Σ → K be the canonical projection.Then dim H K is the unique root of P π tψ 0.
First we show P π tψ is decreased with respect to t.If 0 ≤ t 1 ≤ t 2 , then for any m ∈ M σ Σ , we have h π σ, m t 1 ψ dm ≥ h π σ, m t 2 ψ dm.Hence according to variational principle, we have P π t 1 ψ ≥ P π t 2 ψ .
Let a 1 , a 2 , . . ., a k be given real numbers.If p i ≥ 0 and k i 0 p i 1, then k i 0 p i a i − log p i ≤ log k d .Assume that #{1 ≤ i ≤ l : x ∈ S i K } ≤ k 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.
|v| 1 .Definition 3.2.The IFS {S i } l i 1 is conformal if for every i ∈ {1, 2, . . ., l}, 1 S i : U → S i U is C 1 , 2 S i x / 0 for all x ∈ U, and 3 |S i x y| S i x |y| for all x ∈ U, y ∈ R d .,...,x n πσ n x denote the differential of S x 1 ,...,x n : S x 1 • S x 2 • • • • • S x n at πσ n x.When λ x λ x , the common value, denoted as λ x , is called the Lyapunov exponents of {S i } l i 1 at x.It is easy to check that both λ and λ are positive-valued σ-invariant functions on Σ i.e., λ λ • σ and λ λ • σ .Definition 3.4.A C 1 IFS {S i } l i 1 is said to be weakly conformal if 1 n log S x 1 ,...,x n πσ n x − log S x 1 ,...,x n πσ n x 3.3converges to 0 uniformly on Σ as n tends to ∞.