The asymptotically additive topological pressure on the irregular set for asymptotically additive potentials

In this note we prove that every weak Gibbs measure for an asymptotically additive sequences is a Gibbs measure for another asymptotically additive sequence. In particular, a weak Gibbs measure for a continuous potential is a Gibbs measure for an asymptotically additive sequence. This allows, for example, to apply recent results on dimension theory of asymptotically additive sequences to study multifractal analysis for weak Gibbs measure for continuous potentials.

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“…In order to prove proposition 1.2 we require some auxiliary results. A version of the following Lemma was proved in [ZZC,Lemma 2.3].…”

Section: Introductionmentioning

confidence: 99%

Weak Gibbs measures as Gibbs measures for asymptotically additive sequences

Iommi

Yayama²

2016

Proc. Amer. Math. Soc.

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“…Moreover, the previous result holds for uniformly expanding repellers with respect to some C 1 -map on a compact manifold since these admit finite Markov partitions and can be semi-conjugate to subshifts of finite type. [39], the pressure of Λ is defined by the limit:…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

“…Morever, if Φ = {ϕ n } is almost additive then it is asymptotically additive (see e.g. [39]). By Kingman's subadditive ergodic theorem it follows that for every sub-additive sequence Φ = {ϕ n } and every f -invariant ergodic probability measure µ so that ϕ 1 ∈ L 1 (µ) it holds lim n→∞ 1 n ϕ n (x) = inf n≥1 1 n ϕ n dµ =: F * (Φ, µ), for µ-a.e.…”

Section: Introductionmentioning

confidence: 99%

Multifractal analysis of the irregular set for almost-additive sequences via large deviations

Bomfim

Varandas

2015

Nonlinearity

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In this paper we introduce a notion of free energy and large deviations rate function for asymptotically additive sequences of potentials via an approximation method by families of continuous potentials. We provide estimates for the topological pressure of the set of points whose non-additive sequences are far from the limit described through Kingman's sub-additive ergodic theorem and give some applications in the context of Lyapunov exponents for diffeomorphisms and cocycles, and Shannon-McMillan-Breiman theorem for Gibbs measures.

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“…It is not hard to see that an almost additive potential is asymptotically additive, see [19,Proposition A.5] or [38,Proposition 2.1] for proofs. Hence, Theorem A-F are true for any asymptotically additive potentials.…”

Section: Examplesmentioning

confidence: 99%

“…In [15], author studied the sub-growth rate of asymptotically sub-additive potentials and subordination principle for sub-additive potentials. One of the motivation of the present paper is to study the ergodic optimization for a particular sequence of asymptotically additive potentials (see precise definition in the next section and use AAP for short) which arises naturally in the study of the dimension theory in dynamical systems (see [19,38] for examples of AAP). The study of this paper is also motivated by the theory of multifractal analysis.…”

Section: Introductionmentioning

confidence: 99%