Abstract

Let be a weakly conformal iterated function system on with attractor . Let be the canonical projection. In this paper we define a new concept called “projection pressure” for 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 .

1. Introduction

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

There are many references to compute the Hausdorff dimension of 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 , especially the Bowen’s equation. Usually, we call the Bowen’s equation, where is the topological pressure of the map , and is the geometric potential . 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 whenever is a conformal map on a Riemannian manifold and 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 , then the pressure function has a unique zero root . In 2000, using the definition of Carathe’ odory dimension characteristics, Barreira and Schmeling [11] introduced the notion of the -dimension for positive functions , showing that is the unique number such that .

On the progress of calculating , [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 -Cantor set [12].

In order to study the Hausdorff dimension of an invariant measure 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 . Denote by its attractor. Let associated with the left shift . Let denote the space of -invariant measure on , endowed with the weak-star topology, the space of real-valued continuous functions of , and be the canonical projection defined by

A measure on is called invariant (resp., ergodic) for the IFS if there is and invariant (resp., ergodic) measure on such that .

Let be a probability space. For a sub--algebra of and , we denote by the conditional expectation of given . For countable -measurable partition of . We denote by the conditional information of given , which is given by the formular: where denote the characteristic function on .

The conditional entropy of given , written is defined by the formula .

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

Now we consider the space , where is the Borel -algebra on and . Let denote the Borel partition: of , where . Let denotes the -algebra:

For convenience, we use to denote the Borel -algebra of . For , denote and , for all . Let .

Definition 2.1. For any , we call the projection entropy of under w.r.t , and we call the local projection entropy of at under w.r.t , where denote the function .

It is clear that .

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 , one has , where denotes the classical measure-theoretic entropy of associated with .(ii)The map is affine on . Furthermore if is the ergodic decomposition of , one has (iii)For any , one has for m-a.e. , where denotes the local entropy of at , that is, .

Definition 2.3. Let and . Define

The term can be viewed as the projection measure-theoretic entropy of w.r.t. the IFS . The following lemma exploits the connection between and , where .

Lemma 2.4. Let and . Set . Then is -invariant, and .

Proof. See Proposition  4.3 in [13].

Definition 2.5. An IFS on a compact set is said to satisfy the asymptotically weak separation condition (AWSC), if where is given by here is the attractor of .

Lemma 2.6. Let be an IFS with attractor . Suppose that is a subset of such that there is a map : so that Let denote the one-side full shift over . Define : by . Then(i) is also the attractor of . Moreover, if one lets denote the canonical projection w.r.t. , then one has .(ii)Let . Then . Furthermore, .

Proof. See Lemma  4.23 in [13].

Lemma 2.7. Let be an IFS with attractor. Assume that for some and each . Then for any .

Proof. See Lemma 4.21 in [13].

Lemma 2.8. Let be given real numbers. If and , then and equality holds iff .

Proof. See Lemma  9.9 in [17].

For convenience, for , write . According to Lemma 2.6 there is a set and a map such that for . Let denote the one-sided full shift space over the alphabet and denote the natural generator. Let be defined by

Theorem 2.9. Suppose an IFS satisfies the AWSC with attractor and is continuous. Then

Proof. We divided the proof into two steps.
Step 1. For arbitrary , then . By Lemma 2.8, we have By Lemma 2.2(i) and Lemma 2.6(ii), divided by yields By the arbitrariness of and , we have Step 1.
Step 2. By the continuity of , for arbitrary , there exists such that for arbitrary and any , we have Now, for any and
Define a Bernoulli measure on by Then can be viewed as a -invariant measure on (by viewing as a subset of ). By Lemma 2.6, we have . Define . We have Let and let , then . We have Since is arbitrary, we finish the proof of Step 2.

Definition 2.10. If an IFS satisfies AWSC with attractor and . We call the projection pressure of under w.r.t. .

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

Corollary 2.11. .

3. Application for Projection Pressure

Definition 3.1. is called a IFS on a compact set if each extends to a contracting -diffeomorphism on an open set .

For any real matrix , we use to denote the usual norm of and the smallest singular value of , that is,

Definition 3.2. The IFS is conformal if for every , (1) is , (2) for all , and (3) for all , .

Definition 3.3. Let be a IFS. For , the upper and lower Lyapunov exponents of at are defined, respectively, by where denote the differential of at . When , the common value, denoted as , is called the Lyapunov exponents of at .
It is easy to check that both and are positive-valued -invariant functions on (i.e., and ).

Definition 3.4. A IFS is said to be weakly conformal if converges to 0 uniformly on as 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