Differentiability of thermodynamical quantities in non-uniformly expanding dynamics

Bomfim, Thiago; Castro, Armando; Varandas, Paulo

doi:10.48550/arxiv.1205.5361

Cited by 3 publications

(8 citation statements)

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“…where S n ψ = n−1 j=0 ψ • f j is the usual Birkhoff sum, can often be proved to satisfy E f,φ,ψ (t) = P top (f, φ + tψ) − P top (f, φ) (see e.g. [15,7]).…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

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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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“…where S n ψ = n−1 j=0 ψ • f j is the usual Birkhoff sum, can often be proved to satisfy E f,φ,ψ (t) = P top (f, φ + tψ) − P top (f, φ) (see e.g. [15,7]).…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

“…for every small ε ∈ (0, 1) and t ∈ R (see e.g. [15,7]). Using this variational property we prove in Section 3.3 that it is possible define the Legendre transform of the corresponding free energy functions of Ψ as…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

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

Bomfim

Varandas

2015

Nonlinearity

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“…The interval J may depend on f , φ and ψ and that I f,φ,ψ (s) can often be proved to be a (local) level-1 large deviations rate function (see e.g. [39,31,3]…”

Section: 2mentioning

confidence: 99%

“…One should also point out that if a local large deviations principle holds as in equations (2.3) and (2.4) (e.g. [39,31,3]) then it is not hard to see that an upper bound for P X c (f, φ) can be taken as…”

Section: 2mentioning

confidence: 99%

“…There are examples where the right hand side term above can also be shown to vary continuously with the data even in the non-uniformly expanding context (see e.g. [3]). Since we only estimated the topological pressure of the sets X c in the non-uniformly expanding context one question that arises naturally is the following Question: Are there examples of transitive non-uniformly expanding maps under the conditions of the previous theorem where P Xc (f, φ) differs from P X c (f, φ) and coincides with the topological pressure P top (f, φ)?…”

Section: 2mentioning

confidence: 99%

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Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets

Bomfim¹,

Varandas²

2015

Ergod. Th. Dynam. Sys.

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Abstract. In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if f has an expanding repeller and φ is an Hölder continuous potential we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval I ⊂ R can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we deduce that most irregular sets for maps with the specification property have topological pressure strictly smaller than the whole system. Some extensions to a non-uniformly hyperbolic setting, level-2 irregular sets and hyperbolic flows are also given.

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Contraction in the Wasserstein metric for some Markov chains, and applications to the dynamics of expanding maps

2015

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Abstract. We employ techniques from optimal transport in order to prove decay of transfer operators associated to iterated functions systems and expanding maps, giving rise to a new proof without requiring a Doeblin-Fortet (or Lasota-Yorke) inequality.Our main result is the following. Suppose T is an expanding transformation acting on a compact metric space M and A : M → R a given fixed Hölder function, and denote by L the Ruelle operator associated to A. We show that if L is normalized (i.e. if L (1) = 1), then the dual transfer operator L * is an exponential contraction on the set of probability measures on M with the 1-Wasserstein metric.Our approach is flexible and extends to a relatively general setting, which we name Iterated Contraction Systems. We also derive from our main result several dynamical consequences; for example we show that Gibbs measures depends in a Lipschitzcontinuous way on variations of the potential.

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Differentiability of thermodynamical quantities in non-uniformly expanding dynamics

Cited by 3 publications

References 0 publications

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

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

Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets

Contraction in the Wasserstein metric for some Markov chains, and applications to the dynamics of expanding maps

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