Received 14 September 2014; revised 15 October 2014; accepted 21 October 2014

ABSTRACT

Multifractal analysis studies level sets of asymptotically defined quantities in dynamical systems. In this paper, we consider the -dimension spectra on such level sets and establish a conditional variational principle for general asymptotically additive potentials by requiring only existence and uniqueness of equilibrium states for a dense subspace of potential functions.

Keywords:

Multifractal Analysis, -Dimension Spectra, Asymptotically Additive

1. Introduction

The theory of multifractal analysis is a subfield of the dimension theory in dynamical systems. A general framework for multifractal analysis of dynamical systems was laid out in [1] [2] . It studies a global dimensional quantity that assigns to each level set a “size” or “complexity”, such as its topological entropy or Hausdorff dimension. Broadly speaking, let be a continuous transformation of a compact metric space; let, be potential functions defined on with value in. Given, we consider the level set:

The dimension spectrum (of potential) is defined by which has been extensively studied for Hólder continuous potentials for conformal repellers in [3] - [5] .

In [6] , Barreira, Saussol, and Schmeling extended their work to higher-dimensional multifractal spectra, moreover, for which they consider the more general -dimension in place of the topological entropy. Precisely, they consider functions, with and examine the level sets

for. We denote by the family of -invariant Borel probability measures on, and define a continuous function:

Given a positive function we denote by the -dimension of the set (see Section 2 for the definition). Let be the family of continuous functions with a unique equilibrium measure, they obtain the following result:

Theorem 1. Assume that the metric entropy of is upper semi-continuous, and that .

If,. Otherwise, if, , and the following properties hold:

(I) satisfies the variational principle:

(II), where is the unique real number satisfying:

(III) There exists ergodic measure with and such that

In [7] , Barreira and Doutor study the spectrum of the -dimension for the class of almost additive sequences with a unique equilibrium measure and establish a conditional variational principle for the dimension spectra in the context of the nonadditive thermodynamic formalism. We recall that a sequence of functions is said to be almost additive (with respect to a transformation) if there is a constant such that for every, we have:

In [8] Climenhaga proved a generalisation of Theorem 1 provided that there is a dense subspace of comprising potentials with unique equilibrium states, i.e., the result applies to all continuous functions, not just those whose span lies inside the collection of potentials with unique equilibrium states.

This paper is devoted to the study of higher-dimensional multifractal analysis for the class of asymptotically additive potentials. We consider the multifractal behavior of -dimension spectrum of level sets and establish the conditional variational principle under the assumption proposed by Climenhaga.

Section 2 gives definitions and notions, and Section 3 gives precise formulations of the result and proofs.

2. Preliminaries

We recall in this section some notions and results from the thermodynamic formalism.

2.1. Nonadditive Topological Pressure

We first introduce the notion of nonadditive topological pressure. We also refer the reader to [2] and [7] for further references.

Let be a continuous transformation of a compact metric space. We denote by the space of continuous functions on and the set of all -invariant measures. Given a finite open cover of, we denote by the collection of vectors with. For each, we write, and we consider the open set

Now let be a sequence of continuous functions. For each we define:

We always assume that

(1)

For each we write:

(2)

Given a set and, we define the function:

where the infimum is taken over all finite or countable collections, such that. We also define

It was shown in [9] that the limit

exists. The number is called the nonadditive topological pressure of in the set (with respect to ). In particular, if, we get the topological entropy. We also write.

The following proposition was established in [2] .

Proposition 1. For any, we have

2.2. -Dimension

We recall here a notion introduced by Barreira and Schmeling in [10] . Let be a strictly positive continuous function. Likewise, we define

where is defined as in (2) and where the infimum is taken over all finite or countable collections such that. We also define

Theorem 2. ( [10] ) The following limits exist:

We call the -dimension of. If, then the number coincides with the topological entropy of on. The following result is an easy consequence of the definitions.

Proposition 2. The number is the unique root of the equation, where with for each.

Furthermore, given a probability measure in, we set:

We can show that the limit exists, and we call it the -dimension of. When is ergodic, one can show that (see [10] )

(3)

2.3. Asymptotically Additive Sequences

This kind of potential was introduced by Feng and Huang ( [11] ).

Definition 1. A sequence of functions on is said to be asymptotically additive if for any, there exists such that

We denote by the family of asymptotically additive sequences of continuous functions (satisfying (1)). Now we give two propositions whose proof can be found in [11] .

Proposition 3. If is a continuous transformation of a compact metric space, is an asymptotically additive sequence, and, then

(I) The limit exists for;

(II) The limit exists;

(III) If is ergodic, then for ,

(4)

(IV) The function is continuous with the weak* topology in.

Proposition 4. If is a continuous transformation of a compact metric space, is an asymptotically additive sequence, then the topological pressure satisfies the following variational principle:

We call an equilibrium measure for the potential if

Note that if the function is upper semicontinuous, then every sequence in has an equilibrium measure.

3. Main Result

Let and take. We write and, and also,.

We assume that

(1) There exists constant such that for any and any.

(2) For every, for and every, where the limit exists by proposition 3.

Given, we define:

and function by.

We also consider the function defined by:

Given vectors and we use the notations:

and

We also consider the positive sequence of functions with.

Our main result is the following theorem.

Theorem 3. Let be a continuous transformation of a compact metric space such that the entropy map is upper semicontinuous, and assume that there exists a dense subset such that every has a unique equilibrium measure.

If, then. Otherwise, if, then, and the following properties hold:

(I) satisfies the variational principle:

(II), where is the unique real number satisfying:

(III) There exists ergodic measure with, , and

which is arbitrarily close to.

Proof. We first establish several auxiliary results.

Lemma 1. For there exists constant such that for every we have

(5)

where denotes the supremum norm.

Proof. For any, since the sequence is asymptotically additive, there exists such that

Therefore, there exists, such that for every and, we have

and thus

□

Lemma 2. If, then.

Proof. Using (5), a slight modification of the proof of Lemma 2 in [7] yields this statement, and thus we omit it. □

Lemma 3. If, then. Otherwise, if, then.

Proof. Take with and let. Then for. We consider the sequence of probability measures in defined by. Let be a limitpoint of, clearly. We always assume is ergodic, or else taking an ergodic decomposition of. The desired statements are thus immediate consequences of (4). □

Now proceed with the proof of (1) in theorem 3. We use analogous arguments to those in the proof of lemma 3 in [7] . First show that

Let be the distance of to. Take and define:

Given with for each, we have:

and hence. Therefore, there exists such that

where,. Moreover,

Since

we obtain:

Since, it follows that

It implies that takes arbitrarily large values for sufficiently large, and hence there exists such that for every with. The continuity of implies that it attains a minimum at some point with.

Note that is a dense subset such that every has a unique equilibrium measure, then for every and there exists, and with the following properties:

(1) has a unique equilibrium measure which depends continuously on (for fixed);

(2);

(3).

Therefore,

and thus

Denote a limit point of as, then

(6)

For each vector with, let and let be taken as in (6). We have

If, then

Now assume that when for some measure. The upper semicontinuity of the entropy implies that

This shows that is an equilibrium measure of

satisfying

Similarly, one can consider and find an invariant measure that is an equilibrium measure of satisfying

For each, let. Then the function

is continuous. Moreover, and. Hence, there exists such that. Since and are equilibrium measures of, this implies that is also an equilibrium measure of. Therefore, for each unit vector there exists an equilibrium measure of such that

(7)

We claim that there exists an equilibrium measure of such that

(8)

Let us assume that such a measure does not exist. We denote by the set of all equilibrium measures of. Then

is a compact convex subset of. Hence, there exist a unit vector and such that for.

For every, we have

which contradicts (7). This completes the proof of claim. Observe that this claim implies.

By lemma 2, for the measure satisfying (8), we have

and hence

We now to prove the reverse inequality. We need the following lemma.

Lemma 4. ([8] ) Under the assumptions of theorem 3, for and, we have:

In fact, this is a particular case of Theorem C in [8] .

For any, there exists with such that. Therefore

and hence by proposition 2 we have. The arbitrariness of implies that

and thus

(9)

Furthermore, since the map is upper semicontinuous on the compact set, then the supremum of (9) can be obtained, i.e.

This completes (I) of theorem 3.

We now proceed with the proof of (II) and (III). By lemma 2 we have

for every. Therefore,

On the other hand, for any we have

and hence. So we conclude that

By ergodic decomposition we obtain

For any, there exists ergodic with such that

Note that, then by (3)

It follows that statement (III) in theorem 3 holds. □

Acknowledgements

The author wishes to thank Professor Cao Yongluo for his invaluable suggestions and encouragement.

References