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

Kloeckner, Benoît; Lopes, Artur O.; Stadlbauer, Manuel

doi:10.1088/0951-7715/28/11/4117

Cited by 17 publications

(21 citation statements)

References 22 publications

(27 reference statements)

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“…Recall that a path in a metric space defined on an interval I is said to be absolutely continuous when it has a metric speed in L 1 (I): this is a very weak regularity, so that Theorem E can be interpreted as meaning that a small perturbation of the potential induces a brutal reallocation of the mass distribution in the sense of W 2 . This contrasts with a Lipschitz regularity result obtained for the 1-Wasserstein metric in [KLS14] (but note that W 1 does not yield a differential structure).…”

Section: From Integral Differentiation To a Riemannian Metriccontrasting

confidence: 96%

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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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Section: From Integral Differentiation To a Riemannian Metriccontrasting

confidence: 96%

“…Example 2.6. When T is expanding in a relatively general sense and X (Ω) is a space of Hölder functions, (H3) and (H4) are proved in [KLS14] (the spectral gap is proved there for normalized potentials only, but see remark 2.9).…”

Section: Transfer Operatormentioning

confidence: 99%

The calculus of thermodynamical formalism

et al. 2018

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“…Observe that d r ≤d ≤ αd r which implies that W r ≤W ≤ αW r andD ≤ D r ≤ αD. The following Theorem is an adaption of Lemma 2.1 in [10] (see also [8]). Theorem 2.2.…”

Section: Exponential Decay and Continuitymentioning

confidence: 85%

“…The authors would like to point out that the novelty of this section is the method of proof based on the Vaserstein metric and the family {P x m,n }. The arguments for the contraction of the Vaserstein metric were already known in the context of Markov operators (see [6]) and for normalized potentials ( [10,8]). By considering {P x m,n }, these arguments immediately can be adapted to arbitrary Hölder potentials without a priori having knowledge about the properties of {h x }, {ρ x } and {ν x }.…”

Section: Exponential Decay and Continuitymentioning

confidence: 99%

On the Lyapunov spectrum of relative transfer operators

Bessa

Stadlbauer

2016

Stoch. Dyn.

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We analyze the Lyapunov spectrum of the relative Ruelle operator associated with a skew product whose base is an ergodic automorphism and whose fibers are full shifts. We prove that these operators can be approximated in the $C^0$-topology by positive matrices with an associated dominated splitting.Comment: The article now contains a section on decay of correlations of relative transfer operator

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“…Under certain assumptions, we can compute a formula forL. In fact we have the following (see Propositions 28,35,34).…”

Section: A General Linear Response Results For Regularizing Markov Opementioning

confidence: 99%

A linear response for dynamical systems with additive noise

Galatolo

Giulietti

2019

Nonlinearity

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We show a linear response statement for fixed points of a family of Markov operators which are perturbations of mixing and regularizing operators. We apply the statement to random dynamical systems on the interval given by a deterministic map T with additive noise (distributed according to a bounded variation kernel). We prove linear response for these systems, also providing explicit formulas both for deterministic perturbations of the map T and for changes in the noise kernel. The response holds with mild assumptions on the system, allowing the map T to have critical points, contracting and expanding regions. We apply our theory to topological mixing maps with additive noise, to a model of the Belozuv-Zhabotinsky chemical reaction and to random rotations. In the final part of the paper we discuss the linear request problem for these kind of systems, determining which perturbations of T produce a prescribed response.

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

Cited by 17 publications

References 22 publications

The calculus of thermodynamical formalism

The calculus of thermodynamical formalism

On the Lyapunov spectrum of relative transfer operators

A linear response for dynamical systems with additive noise

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