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

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 [

The dimension spectrum

In [

they consider functions

for

Given a positive function

Theorem 1. Assume that the metric entropy of

If

(I)

(II)

(III) There exists ergodic measure

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

In [

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

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

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

We first introduce the notion of nonadditive topological pressure. We also refer the reader to [

Let

Now let

We always assume that

For each

Given a set

where the infimum is taken over all finite or countable collections

It was shown in [

exists. The number

The following proposition was established in [

Proposition 1. For any

We recall here a notion introduced by Barreira and Schmeling in [

where

Theorem 2. ( [

We call

Proposition 2. The number

Furthermore, given a probability measure

We can show that the limit

This kind of potential was introduced by Feng and Huang ( [

Definition 1. A sequence

We denote by

Proposition 3. If

(I) The limit

(II) The limit

(III) If

(IV) The function

Proposition 4. If

is an asymptotically additive sequence, then the topological pressure

We call

Note that if the function

Let

We assume that

(1) There exists constant

(2) For every

Given

and function

We also consider the function

Given vectors

and

We also consider the positive sequence of functions

Our main result is the following theorem.

Theorem 3. Let

every

If

properties hold:

(I)

(II)

(III) There exists ergodic measure

which is arbitrarily close to

Proof. We first establish several auxiliary results.

Lemma 1. For

where

Proof. For any

Therefore, there exists

and thus

Lemma 2. If

Proof. Using (5), a slight modification of the proof of Lemma 2 in [

Lemma 3. If

Proof. Take

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

Let

Given

and hence

where

Since

we obtain:

Since

It implies that

Note that

(1)

(2)

(3)

Therefore,

and thus

Denote

For each vector

If

Now assume that

This shows that

satisfying

Similarly, one can consider

For each

is continuous. Moreover,

We claim that there exists an equilibrium measure

Let us assume that such a measure does not exist. We denote by

is a compact convex subset of

For every

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

By lemma 2, for the measure

and hence

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

Lemma 4. ([

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

For any

and hence by proposition 2 we have

and thus

Furthermore, since the map

This completes (I) of theorem 3.

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

for every

On the other hand, for any

and hence

By ergodic decomposition we obtain

For any

Note that

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

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