Almost sure rates of mixing for i.i.d. unimodal maps

Baladi, Viviane; Benedicks, Michael; Maume-Deschamps, Véronique

doi:10.1016/s0012-9593(01)01083-7

Cited by 47 publications

(94 citation statements)

References 21 publications

(54 reference statements)

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“…Our choice is motivated by the fact that our main result for the extreme values in presence of noise could be relatively easily shown with those assumptions, but it is also clear from the proof where possible generalizations could occur and we will quote a few of them. Let us notice that other authors basically used additive noise when they studied statistical properties of random dynamical systems [235,236,237], for instance. Random transformations with additive noise will be investigated with two methods: the first is the usual probabilistic approach which is at the base of all the previous mathematical formalisations presented in this book: we will briefly refer to it as the probabilistic approach.…”

Section: Proposition 6113 Assume That P : σ → σ Is the Lorenz Map mentioning

confidence: 99%

Extremes and Recurrence in Dynamical Systems

Lucarini¹,

Faranda²,

Freitas³

et al. 2016

189

253

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ii 4.1.2 The New Conditions 44 4.1.3 The Existence of EVL for General Stationary Stochastic Processes under Weaker Hypotheses 46 4.1.4 Proofs of Theorem 4.1.4 and Corollary 4.1.5 48 4.2 Extreme Values for Dynamically Defined Stochastic Processes 53 4.2.1 Observables and Corresponding Extreme Value Laws 55 4.2.2 Extreme Value Laws for Uniformly Expanding Systems 59 4.2.3 Example 4.2.1 revisited 61 4.2.4 Proof of the Dichotomy for Uniformly Expanding Maps 63 4.3 Point Processes of Rare Events 64 4.3.1 Absence of Clustering 64 4.3.2 Presence of Clustering 65 4.3.3 Dichotomy for Uniformly Expanding Systems for Point Processes 67 4.4 Conditions Д q (u n ), D 3 (u n ), D p (u n ) * and Decay of Correlations 68 4.5 Specific Dynamical Systems where the Dichotomy Applies 71 4.5.1 Rychlik Systems 72 4.5.2 Piecewise Expanding Maps in Higher Dimensions 73 4.6 Extreme Value Laws for Physical Observables 74

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Section: Proposition 6113 Assume That P : σ → σ Is the Lorenz Map mentioning

confidence: 99%

Extremes and Recurrence in Dynamical Systems

Lucarini¹,

Faranda²,

Freitas³

et al. 2016

189

253

View full text Add to dashboard Cite

show abstract

“…(In view of the results by [13] mentioned above, one can expect the correlations of (5) to decay at least stretched exponentially. )…”

Section: Low Dimensionsmentioning

confidence: 96%

“…With Benedicks and Maume-Deschamps [13], we consider a good unimodal map f (satisfying Benedicks-Carleson type assumptions), and for small > 0 a probability ν on Ω 0 = [− , ], setting f ω (x) = f (x)+ω 0 . Assuming e.g.…”

Section: Small Random Perturbations and Stochastic Stabilitymentioning

confidence: 99%

Spectrum and Statistical Properties of Chaotic Dynamics

Baladi

2001

European Congress of Mathematics

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Abstract. We present new developments on the statistical properties of chaotic dynamical systems. We concentrate on the existence of an ergodic physical (SRB) invariant measure and its mixing properties, in particular decay of its correlation functions for smooth observables. In many cases, there is a connection (via the spectrum of a Ruelle-Perron-Frobenius transfer operator) with the analytic properties of a weighted dynamical zeta function, weighted dynamical Lefschetz function, or dynamical Ruelle-Fredholm determinant, built using the periodic orbit structure of the map.

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“…the past random orbit of X at ω (see [2] for this notation). We say that X has historic behaviour at ω if there exist real numbers α, β and a continuous function ϕ :…”

Section: The Proofmentioning

confidence: 99%

Historic Behaviour for Random Expanding Maps on the Circle

Nakano¹

2017

Tokyo J. Math.

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Takens constructed a residual subset of the state space consisting of initial points with historic behaviour for expanding maps on the circle. We prove that this statistical property of expanding maps on the circle is preserved under small random perturbations. The proof is given by establishing a random Markov partition, which follows from a random version of Shub's Theorem on topological conjugacy with the folding maps.

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Almost sure rates of mixing for i.i.d. unimodal maps

Abstract: ABSTRACT. -It has been known since the pioneering work of Jakobson and subsequent work by Benedicks and Carleson and others that a positive measure set of quadratic maps admit an absolutely continuous invariant measure.

Cited by 47 publications

References 21 publications

Extremes and Recurrence in Dynamical Systems

Extremes and Recurrence in Dynamical Systems

Spectrum and Statistical Properties of Chaotic Dynamics

Historic Behaviour for Random Expanding Maps on the Circle

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