Dynamics of random transformations: smooth ergodic theory

Liu, Peidong

doi:10.1017/s0143385701001614

Cited by 42 publications

(44 citation statements)

References 103 publications

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“…Then the entropy of η is equal to the integral of entropies of its ergodic components (see [9], page 1289 and references there in), of course the same occurs with the Ψ(η) (*). If (x, w) / ∈ {(x, w); η (x,w) ∈ K α } then by lemma 5.14:…”

Section: Entropy Lemmasmentioning

confidence: 93%

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Equilibrium states for random non-uniformly expanding maps

2004

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Abstract. We show that, for a robust (C 2 -open) class of random nonuniformly expanding maps, there exists equilibrium states for a large class of potentials.In particular, these sytems have measures of maximal entropy. These results also give a partial answer to a question posed by Liu-Zhao. The proof of the main result uses an extension of techniques in recent works by Alves-Araújo, Alves-Bonatti-Viana and Oliveira.

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Section: Entropy Lemmasmentioning

confidence: 93%

“…Entropy We follow Liu [9] on the definition of the Kolmogorov-Sinai entropy for random transformations:…”

Section: Random Transformations and Invariant Measuresmentioning

confidence: 99%

Equilibrium states for random non-uniformly expanding maps

2004

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“…For the deterministic case we refer to [27], [26], [22] [18] and [19], and for the random case, we refer to [2], [3], [20] and [21]. It gives an explicit formula relating the measure-theoretic entropy and Lyapunov exponents in corresponding settings.…”

Section: Measure-theoretic Entropy Of Random Z K -Actionsmentioning

confidence: 99%

Entropy formula for random ℤ^{𝕜}-actions

Zhu

2016

Trans. Amer. Math. Soc.

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In this paper, entropies, including measure-theoretic entropy and topological entropy, are considered for random Z k -actions which are generated by random compositions of the generators of Z k -actions. Applying Pesin's theory for commutative diffeomorphisms we obtain a measure-theoretic entropy formula of C 2 random Z k -actions via the Lyapunov spectra of the generators. Some formulas and bounds of topological entropy for certain random Z k (or Z k + )-actions generated by more general maps, such as Lipschitz maps, continuous maps on finite graphs and C 1 expanding maps, are also obtained. Moreover, as an application, we give a formula of Friedland's entropy for certain C 2 Z k -actions.

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“…Random dynamical systems are nonuniform in nature in terms of hyperbolicity. There is an extensive literature on the nonuniformly hyperbolic theory and the ergodic theory for both independent and identically distributed random transformations and stationary random dynamical systems, which we refer to Arnold [4], Kifer [32,34,33], Ledrappier and Young [37,38], Liu and Qian [44], Liu [43], Kifer and Liu [35], and the references therein.…”

Section: Intoductionmentioning

confidence: 99%

Entropy, Chaos and Weak Horseshoe for Infinite Dimensional Random Dynamical Systems

Huang

2013

XVIIth International Congress on Mathematical Physics

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Abstract. In this paper, we study the complicated dynamics of infinite dimensional random dynamical systems which include deterministic dynamical systems as their special cases in a Polish space. Without assuming any hyperbolicity, we proved if a continuous random map has a positive topological entropy, then it contains a topological horseshoe. We also show that the positive topological entropy implies the chaos in the sense of Li-Yorke. The complicated behavior exhibiting here is induced by the positive entropy but not the randomness of the system.

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Dynamics of random transformations: smooth ergodic theory

Abstract: A selection of basic results in smooth ergodic theory and in the thermodynamic formalism of dynamical systems generated by compositions of random maps is reviewed in this article.

Cited by 42 publications

References 103 publications

Equilibrium states for random non-uniformly expanding maps

Equilibrium states for random non-uniformly expanding maps

Entropy formula for random ℤ^{𝕜}-actions

Entropy, Chaos and Weak Horseshoe for Infinite Dimensional Random Dynamical Systems

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