Multifractal analysis on the historic set in non-uniformly expanding systems

Introduction

This paper is devoted to the study of the historic set of asymptotically additive potential in a class of one dimensional non-uniformly hyperbolic systems. Before formulating our results, we first present some background and motivation.

Historic set

We call a topological dynamical system (for short TDS) if is a compact metric space and is a continuous map. We denote the set of continuous functions from to R by C(X, R). For any given f ∈ C(X, R), let and be the Birkhoff sum and the Birkhoff average of respectively. We call the orbit { } of having "historical behavior" if the limit does not exists for some f ∈ C(X, R). David. Ruelle firstly introduced this term in [18] . To pay tribute to Ruelle, we call a point historic point if the orbit of has historic behavior and call the set of all historic points as historic set. Flois. Takens discuss historic set further more in [19] and proposed the following problem: whether there are persistent classes of smooth dynamical systems such that the set initial states which give rise to orbits with historic behavior has positive lebesgue measure. This problem is called "Takens last problem", see [12,13,14,5] .

We call poin "generic point" if the limit exists for each φ ∈ C(X, R). Limit does not exists means that as time goes to , " " keeps having new ideas about what it want to do. The behavior of the orbit of "generic point" is predicable. However, the orbit of "historic point" contains more information, it has a history. Ruelle and Takens suggest that "historic set" is more important. For any given f ∈ C(X, R), Let be the historic set of . Then is a subset of historic set and historic set is the union of for all continuous function f ∈ C(X, R).

( , ) XT  X: T X X  X 1 0 :

Recently, the structure of historic set attracts great interests, see [1,2,3,4,6,7,8,9,11,16,17,18,19,20] . By Birkhoff ergodic theory, we have for any given -invariant measure . This implies that historic set is negligible in measure category. However, results of [1] overthrow this viewpoint. In most of the systems, especially uniformly hyperbolic systems, historic set carries full topological entropy and has full Hausdorff dimension. These means that historic set carries most of the information on the system. In finite subshifts and conformal repellers [1,6,8] , authors prove that historic set has full Hausdorff dimension unless it is empty set. In systems with some mild specification property [20] , Daniel. Tompson shows that either historic set carries full topological pressure or it is nothing. In [3] , authors show that historic set is residual if it is not empty set. In systems with specification property [4] , the same authors get a variational principal on the topological entropy of historic set. In this paper, we consider the historic set of asymptotically additive function sequence in a class of non-uniformly expanding systems. We show that the Hausdorff dimension of the historic set in this systems has "dichotomy", i.e. it carries full Hausdorff dimension unless it is empty set. In the following, we introduce the non-uniformly expanding systems considered in this paper.

Non-uniformly expanding systems

We consider the pseudo-Markov maps. Let {: 1, , } i I i m  be the subintervals of the unit interval [0, 1] and 1 :

1] be a piecewise 1 C   map with exponent 0   . Furthermore, we impose the following assumptions:

. We point out coding map  is a bijection except for at most countably many points.

Asymptotically additive potential

Let be a sequence of continuous functions on . By convention of Feng, see [6,8,9] , we call continuous function sequence additive potential if the following quality holds, , for all , , .

It is seen that for additive potential .

We call almost additive potential if the following conditions holds, 1.

is continuous from to , 2. There is a positive constant ()

It is obvious that  is additive if  is almost additive with ( ) 0 C  . The almost additive potential arises naturally in the study on the dimension of invariant set in non-conformal repellers [2] and topological pressure of products of positive matrices [7] .

Φ is called asymptotically additive potential if for any given  >0, there exists g ∈ C(X, R), such that

Lemma A.5 in [9] means that almost additive potential is asymptotically additive potential. However in example 1 in [22], authors give an asymptotically additive potential which is not almost additive. Moreover we can see that asymptotically additive potential arise naturally in the study of dimension theory.

For any given asymptotically additive potential  , let

By asymptotically additivity, the above limit exists. We write

. Now, we give the result of this paper.

Main result

In this paper, we prove that the historic set of the asymptotically additive potential in the systems defined in section 1.

Remark 1 As point out in Section 1.3, almost additive means asymptotically additive, however, asymptotically additive need to be almost additive. Then our result covers the conclusion in [16] .

Preliminaries

In this section, we will introduce some lemmas and give the necessary notations.

Recall {1, 2, , } Am  is the alphabet, A  is the full shift and

. By lemma 1 in [11] and lemma 2.1 in [21] , ()

Let T be the non-uniformly expanding map defined in section 1.2. Then ()  be the image measure of  . The following lemma combined by lemma 2 and lemma 3 in [11] , plays a key role in the proof of Theorem 1.

Lemma 2. For any given ( , )

  M

, there exists a sequence of ergodic  -invariant measures

in the weak star topology and ( , ) ( , )

Next lemma means that the map Φ *: M(Σ, σ) → R is continuous.

Lemma 3.

For any given asymptotically additive potential  , let  be a  -invariant measure and

  H Takes s going to 1, by the arbitrary of ò, We prove Theorem 1. Now, we turn to prove lemma 5.

The idea of proof Lemma 5 is to construct a Moran set M whose image M  sitting in historic set ( ; , ) XT  H such that the Hausdorff dimension of M  satisfies the following inequality, ( , ) ( , ) dim min{ , } ( , ) ( , )

For the convenience of reader, we arrange the proof by 4 steps.

Step 1 Constructing Moran set in odd levels and even levels By lemma 1, we can take real number sequence Var , max | ( ) ( ) | .

Similarly for all 2 ni  , we have 22

Var

By lemma 2 and lemma 3, we can choose  -invariant ergodic measure sequence 2 1 1 ()

Similarly, there exists  -invariant ergodic measure sequence 21 () ,).

For any given 0   , by Ergroff Theorem, there exists (2 1) i      such that 21 ( (2 1))

Inequality (1)-( 8) is necessary in following proof.

Step 2 Constructing Moran set contained in historic set

Using (1), ( 2), ( 4), ( 5), ( 6), (8), by direct computation, we get the following results,

/ 0, and / 1 Let  be the Kolmogrov extension of  to all the borel set of M . It is seen that  is a measure on M and  is supported on M .

Step 4 Estimating the lower bound of Hausdorff dimension of Moran measure

In this step, we will show that for all xM  , the following inequality holds,

Thus, Lemma 5 follows. Now, we go to prove inequality (11). Firstly, we estimate the lower bound of () 1), ( 4), ( 2), ( 5), ( 6), ( 8), we have

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