From rates of mixing to recurrence times via large deviations

Alves, José F.; Freitas, Jorge Milhazes; Luzzatto, Stefano; Vaienti, Sandro

doi:10.1016/j.aim.2011.06.014

Cited by 57 publications

(74 citation statements)

References 40 publications

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“…In dimension one this is not an assumption as it has been shown that a system with a positive Lyapunov exponent and an absolutely continuous invariant measure has a Young tower [153].…”

Section: Non-uniformly Hyperbolic Systemsmentioning

confidence: 99%

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Extremes and Recurrence in Dynamical Systems

Lucarini¹,

Faranda²,

Freitas³

et al. 2016

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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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“…In dimension one this is not an assumption as it has been shown that a system with a positive Lyapunov exponent and an absolutely continuous invariant measure has a Young tower [153].…”

Section: Non-uniformly Hyperbolic Systemsmentioning

confidence: 99%

“…• S2: multidimensional uniformly expanding maps These maps have been extensively investigated in [158,129,142,153,240,241] and we defer to those papers for more details. We consider it a particular case corresponding to smooth boundaries.…”

Section: Proposition 725 If a Random Transformation (Y1) Enjoys Prmentioning

confidence: 99%

Extremes and Recurrence in Dynamical Systems

Lucarini¹,

Faranda²,

Freitas³

et al. 2016

Self Cite

178

253

View full text Add to dashboard Cite

show abstract

“…The subject of large deviations in the context of dynamical systems is, indeed, very important, and the existence of nice rate functions is not generically expected unless trajectories enjoy very good statistical properties [23,24]. In the present case, a number of rigorous results [3,5,25,26], that apply to the Pomeau-Manneville map in the exponential instability case, concern exactly polynomial large deviations. As is crucial in the discussion, let us recall the theorem proved in Ref.…”

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confidence: 81%

Comment on “Lyapunov statistics and mixing rates for intermittent systems”

Artuso

Manchein²

2013

Phys. Rev. E

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In Pires et al. [Phys. Rev. E 84, 066210 (2011)], intermittent maps were considered, and the tight relationship between correlation decay of smooth observables and large deviation estimates as, for instance, employed in Artuso and Manchein [Phys. Rev. E 80, 036210 (2009)], was questioned. We try to clarify the problem and to provide rigorous arguments and an analytic estimate that disprove the objections raised in Pires et al. [Phys. Rev. E 84, 066210 (2011)] when ergodic systems are considered.

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“…In fact, regardless of the rate (in this case n −2 ), as long as it is Downloaded by [Laurentian University] at 04:05 16 September 2013 summable, one can actually show that the system has exponential decay of correlations of Hölder observables against L ∞ (P). See [50,Theorem B].…”

Section: Remarkmentioning

confidence: 98%

Extremal behaviour of chaotic dynamics

Freitas

2013

Dynamical Systems

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We present a review of recent results regarding the existence of extreme value laws for stochastic processes arising from dynamical systems. We gather all the conditions on the dependence structure of stationary stochastic processes in order to obtain both the distributional limit for partial maxima and the convergence of point processes of rare events. We also discuss the existence of clustering which can be detected by an extremal index less than 1 and relate it with the occurrence of rare events around periodic points. We also present the connection between the existence of extreme value laws for certain dynamically defined stationary stochastic processes and the existence of hitting times statistics (or return times statistics). Finally, we make a complete description of the extremal behaviour of expanding and piecewise expanding systems by giving a dichotomy regarding the types of extreme value laws that apply. Namely, we show that around periodic points we have an extremal index less than 1 and at very other point we have an extremal index equal to 1. IntroductionThe purpose of this work is to compile a series of recent results regarding the theory of extreme events for dynamical systems. Extreme means that we are interested in the occurrence of rare but possibly catastrophic events. This motivates its applications to situations where risk assessment is rather crucial.Instead of focusing on means and average behaviour of the system, often ruled by central limit theorems, in this approach, we are interested on the largest (or smallest) observations of sample data in order to understand the extremal behaviour of the system by obtaining corresponding distributional limits.The starting point of our analysis is a stationary stochastic process arising from a chaotic dynamical system simply by evaluating an observable function along the orbits of the system, as explained in Section 3. Then the extremal behaviour is analysed by studying the distributional limit of the partial maximum of such stochastic processes or by investigating the limit of point processes counting the number of exceedances of high thresholds during some time interval.In the context of dynamical systems, the study of extreme value laws (EVL) is a quite recent topic. It appeared first in the pioneering paper of Collet [1], which has been an

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From rates of mixing to recurrence times via large deviations

Cited by 57 publications

References 40 publications

Extremes and Recurrence in Dynamical Systems

Extremes and Recurrence in Dynamical Systems

Comment on “Lyapunov statistics and mixing rates for intermittent systems”

Extremal behaviour of chaotic dynamics

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