Differentiability of thermodynamical quantities in non-uniformly expanding dynamics

Bomfim, Thiago; Castro, Armando; Varandas, Paulo

doi:10.1016/j.aim.2016.01.017

Cited by 38 publications

(49 citation statements)

References 41 publications

(47 reference statements)

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“…Note that Corollary 1.5 deals with the regularity of a natural invariant measure in terms of a varying expanding map, in the same spirit of many previous works (see [Rue98], [BS08], [Bal08], [HM10] and [BCV12]) in which the absolutely invariant measure was considered. These papers are all in the so called Linear Response Theory.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 93%

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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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 93%

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

2015

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“…Moreover, they show that if the topological pressure function depends continuously within such systems, then equilibrium stability holds for Hölder continuous potentials with not very large variation. The subsequent works [7,13] by Bomfim, Castro and Varandas study the continuity and even differentiability of several thermodynamical quantities for certain classes of non-uniformly expanding dynamical systems and potentials with small variation. Here we consider a family of non-uniformly expanding maps and hyperbolic potentials and prove that the topological pressure varies continuously within this family.…”

Section: Introductionmentioning

confidence: 99%

Equilibrium stability for non-uniformly hyperbolic systems

Alves¹,

Ramos²,

Siqueira³

2018

Ergod. Th. Dynam. Sys.

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We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the topological pressure is continuous as a function of the dynamics and the potential. We also prove the existence of finitely many ergodic equilibrium states for non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials. Finally we show that these equilibrium states vary continuously in the weak * topology within such systems.2010 Mathematics Subject Classification. 37A05, 37A35.

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“…At this point we have proved Corollary B from the introduction. Corollaries 3.6 and 3.7 where obtained under different assumptions and with different methods by Bomfim, Castro and Varandas [BCV12]; note that we notably do not assume the high-temperature regime (see their conditions (P) and (P')) and that once our framework is set, our proofs are very simple.…”

Section: Figure 1: Potentials and Gibbs Measuresmentioning

confidence: 94%

The calculus of thermodynamical formalism

et al. 2018

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Given a finite-to-one map T acting on a compact metric space Ω and an appropriate Banach space of functions X (Ω), one classically constructs for each potential A ∈ X a transfer operator L A acting on X (Ω). Under suitable hypotheses, it is well-known that L A has a maximal eigenvalue λ A , has a spectral gap and defines a unique Gibbs measure µ A . Moreover there is a unique normalized potential of the form B := A + f − f • T + c acting as a representative of the class of all potentials defining the same Gibbs measure.The goal of the present article is to study the geometry of the set of normalized potentials N , of the normalization map A → B, and of the Gibbs map A → µ A . We give an easy proof of the fact that N is an analytic submanifold of X and that the normalization map is analytic; we compute the derivative of the Gibbs map; last we endow N with a natural weak Riemannian metric (derived from the asymptotic variance) with respect to which we compute the gradient flow induced by the pressure with respect to a given potential, e.g. the metric entropy functional. We also apply these ideas to recover in a wide setting existence and uniqueness of equilibrium states, possibly under constraints.

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Differentiability of thermodynamical quantities in non-uniformly expanding dynamics

Cited by 38 publications

References 41 publications

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

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

Equilibrium stability for non-uniformly hyperbolic systems

The calculus of thermodynamical formalism

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