Multifractal Analysis of the Asympyotically Additive Potentials

Multifractal analysis studies level sets of asymptotically defined quantities in dynamical systems. In this paper, we consider the u -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.


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 : f X X → be a continuous transformation of a compact metric space; let ( ) be potential functions defined on X with value in  .Given α ∈  , we consider the level set: The dimension spectrum : →    (of potential Φ Ψ ) is defined by  which has been extensively studied for Hólder continuous potentials for 1  C α + 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 u -dimension in place of the topological entropy.Precisely, they consider functions ( ) , , , and examine the level sets ( ) , , , ,  .We denote by ( ) the family of f -invariant Borel probability measures on X , and define a continuous function ( ) ( ) be the family of continuous functions with a unique equilibrium measure, they obtain the following result: Theorem 1. Assume that the metric entropy of f is upper semi-continuous, and that { } ( ) span , , , , , , , , K α ≠ ∅ , and the following properties hold: (I) is the unique real number satisfying: (III) There exists ergodic measure ( ) In [7], Barreira and Doutor study the spectrum of the u -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 ( ) said to be almost additive (with respect to a transformation f ) if there is a constant 0 C > such that for every , n m∈  , we have: In [8] Climenhaga proved a generalisation of Theorem 1 provided that there is a dense subspace of ( ) C X 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 u -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.

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

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 : f X X → be a continuous transformation of a compact metric space.We denote by ( ) C X the space of continuous functions on X and ( ) the set of all f -invariant measures.Given a finite open cover  of X , we denote by , we write ( ) m V n = , and we consider the open set We always assume that For each we write: and α ∈  , we define the function: where the infimum is taken over all finite or countable collections ( ) We also define It was shown in [9] that the limit exists.The number ( ) Z P Φ is called the nonadditive topological pressure of Φ in the set Z (with respect to f ).In particular, if 0 Φ = , we get the topological entropy . We also write ( ) ( ) The following proposition was established in [2].Proposition 1.For any Z X ⊂ , we have

u -Dimension
We recall here a notion introduced by Barreira and Schmeling in [10].Let : u X →  be a strictly positive continuous function.Likewise, we define and where the infimum is taken over all finite or countable collections ( ) We call dim u Z the u -dimension of Z .If 1 u ≡ , then the number dim u Z coincides with the topological entropy of f on Z .The following result is an easy consequence of the definitions.
Proposition 2. The number dim u Z α = is the unique root of the equation , where ( ) Furthermore, given a probability measure µ in X , we set: We can show that the limit diam 0 , dim lim dim exists, and we call it the u -dimension of µ .When ( )

Asymptotically Additive Sequences
This kind of potential was introduced by Feng and Huang ( [11]).Definition 1.A sequence ( ) of functions on X is said to be asymptotically additive if for any 0 ε > , there exists ( ) AA X the family of asymptotically additive sequences of continuous functions (satisfying (1)).Now we give two propositions whose proof can be found in [11].
is a continuous transformation of a compact metric space, ( ) ( ) ∈ is an asymptotically additive sequence, and , then (I) The limit ( ) ( ) exists for . .a e x X µ − ∈ ; (II) The limit (IV) The function ( ) is an asymptotically additive sequence, then the topological pressure ( ) P Φ satisfies the following variational principle: is upper semicontinuous, then every sequence in

( )
AA X has an equilibrium measure.
(2) For every ( ) µ − ∈ and every 1, , i d =  , where the limit exists by proposition 3. Given , ,  , we define: We also consider the function ( ) defined by: ( ) , , , , , , Our main result is the following theorem.Theorem 3. Let f be a continuous transformation of a compact metric space X such that the entropy map ( ) is upper semicontinuous, and assume that there exists a dense subset ( ) , then K α ≠ ∅ , and the following properties hold: is the unique real number satisfying: where ∞ ⋅ denotes the supremum norm.Proof.For any 0 ε > , since the sequence ( ) ψ is asymptotically additive, there exists ( ) Therefore, there exists 0 C > , such that for every m ∈  and x X ∈ , we have Proof.Using ( 5), a slight modification of the proof of Lemma 2 in [7] yields this statement, and thus we omit it.□ Lemma 3. Proof.Take . We always assume µ is ergodic, or else taking an ergodic decomposition of µ .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 . Take d q ∈  and define: for each i , we have: . Therefore, there exists ( ) , , , ( ) we obtain: F q takes arbitrarily large values for q sufficiently large, and hence there exists R ∈  such that ( ) ( ) The continuity of F implies that it attains a minimum at some point 0 q with 0 q R ≤ .
Note that ( ) has a unique equilibrium measure, then for every 0 δ > and ( ) with the following properties: ( ( )    has a unique equilibrium measure q δ µ which depends continuously on q (for fixed δ ); ( For each vector k e ∈  with 1 e = , let 0 q q e ε = + and let q µ be taken as in (6).We have This shows that e ν + is an equilibrium measure of ( ) We claim that there exists an equilibrium measure ν of Ξ such that ( ) Let us assume that such a measure does not exist.We denote by ( ) α  the set of all equilibrium measures of Ξ .Then ( ) ( ) ∫ which contradicts (7).This completes the proof of claim.Observe that this claim implies By lemma 2, for the measure ν satisfying (8), we have We now to prove the reverse inequality.We need the following lemma.In fact, this is a particular case of Theorem C in [8].( This completes (I) of theorem 3. We now proceed with the proof of (II) and (III).By lemma 2 we have We also consider the positive sequence of functions ( ) III) There exists ergodic measure ( ) 0 C > such that for every m ∈  we have ( )

dα∈
 with K α ≠ ∅ and let x K α find an invariant measure e ν − that is an equilibrium measure of Ξ ≥ .Hence, there exists 0 a such that ( )0 0 p a = .Since e ν +and e ν − are equilibrium measures of Ξ , this implies that ( ) e a ν is also an equilibrium measure of Ξ .Therefore, for each unit vector k e ∈  there exists an equilibrium measure ( )
So we conclude that