The Compound Poisson Limit Ruling Periodic Extreme Behaviour of Non-Uniformly Hyperbolic Dynamics

Freitas, Ana Cristina Moreira; Freitas, Jorge Milhazes; Todd, Michael J.

doi:10.1007/s00220-013-1695-0

Cited by 53 publications

(101 citation statements)

References 41 publications

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“…Remark 4.3.6. We remark that the second statement of the previous theorem had already been obtained in [140] but the final version of the dichotomy, which also includes the statement regarding non periodic points was only established in [142].…”

Section: Presence Of Clusteringmentioning

confidence: 79%

“…Then they have been able to pull this information back to the original system, proving the same result for ([0, 1), g α , µ α ). Later in [140], these ideas have been extended to periodic points, ultimately proving the following.…”

Section: Intermittent Mapsmentioning

confidence: 91%

“…Misiurewicz showed in [192] that such maps have an absolutely continuous invariant measure µ. One of the nicest properties of these maps is that one can find an interval X ⊂ [0, 1] such that the first return map to X is a full-branched mapping satisfying the conditions of [160], so using [184,140] we can show an analogue of Theorem 5.5.1 for (I, f λ , µ). Indeed, this type of result, for typical points appeared in [184].…”

Section: Interval Maps With Critical Pointsmentioning

confidence: 99%

“…It first appeared in the pioneering paper of Collet, [72], which has been an inspiration for plenty of the research on this issue. Then the subject has been further addressed and developed in many subsequent contributions including [134,73,135,74,136,137,138,139,49,140,81,46,76,77,78,141,142,79,44].…”

Section: Emergence Of Extreme Value Laws For Dynamical Systemsmentioning

confidence: 99%

“…From the study developed in [140] and as noticed in [142, Appendix B], we can state the following result which applies to general stationary stochastic processes, whose full proof is appearing in the forthcoming paper [154].…”

Section: Presence Of Clusteringmentioning

confidence: 99%

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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: Presence Of Clusteringmentioning

confidence: 79%

Section: Intermittent Mapsmentioning

confidence: 91%