Asymptotics for Return Times of Rank-One Systems

Chaumoître, Vincent; Kupsa, Michal

doi:10.1142/s0219493705001298

Cited by 8 publications

(9 citation statements)

References 13 publications

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“…For example, S = Id works and gives the diagonal 2-tupleF (1) U =F (2) U and the matrix S constructed with all lines equal to the vector t β works too.…”

Section: Stochastic Matrix Of Possible 2-tuples Fmentioning

confidence: 99%

“…Then we apply Theorem 4.1 to obtain a rank-one system (which can be described explicitly) such that along a sequence of neighborhood V n we haveF (2) =F (1) .…”

Section: Implementation Of a Sequence Of Periodic Systems In A Rank-omentioning

confidence: 99%

“…Next in Sec. 3, we concentrate on periodic systems for the class of which we exhibit a stochastic matrix that connectsF (1) U andF (2) U . We also develop some explicit instructive examples showing how this matrix has influence on weak limits of sequences of 2-tuples within the class of periodic systems.…”

Section: Introductionmentioning

confidence: 99%

“…We have 1 − β2 β1 = 1 2 > 0, thus the parameter α runs over [1 2 , 1]. Then, we have λ cannot be equal to 1[1,∞[ which is the standard hitting time asymptotic for rank-one systems[1]. MoreoverFig.…”

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confidence: 99%

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Matrix Technique Representation for Second Return Time Asymptotics

Chaumoître

2006

Stoch. Dyn.

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We answer to questions about n-tuple of return time limit laws raised in [2]. By constructing rank-one systems with a 2-tuple different from (F, F ) along sequence of cylinders, we prove that the second return time limit law is not entirely determined by the first return time limit law. Matrix technique is developed in order to solve the combinatorial problem of 2-tuple return time laws. Thus, using a good implementation of a sequence of periodic systems in a rank-one system, examples of rank-one systems such that the 2-tuple return time limit laws along sequences of cylinders satisfiesF (1) =F (2) are given.

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“…For example, S = Id works and gives the diagonal 2-tupleF (1) U =F (2) U and the matrix S constructed with all lines equal to the vector t β works too.…”

Section: Stochastic Matrix Of Possible 2-tuples Fmentioning

confidence: 99%

“…Then we apply Theorem 4.1 to obtain a rank-one system (which can be described explicitly) such that along a sequence of neighborhood V n we haveF (2) =F (1) .…”

Section: Implementation Of a Sequence Of Periodic Systems In A Rank-omentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Matrix Technique Representation for Second Return Time Asymptotics

Chaumoître

2006

Stoch. Dyn.

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“…There are only a couple of examples of zero-entropy processes, where the limit law is attained along the full sequence of cylinders. These are the process derived from the Fibonacci shift and processes derived from two-column rank-one constructions ( [CK05]). In these cases the limit law for the hitting time is piecewise linear.…”

Section: Introductionmentioning

confidence: 99%

Exponential return times in a zero-entropy process

Grzegorek¹,

Kupsa²

2011

Preprint

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We construct a zero-entropy weakly mixing finite-valued process with the exponential limit law for return resp. hitting times. This limit law is obtained in almost every point, taking the limit along the full sequence of cylinders around the point. Till now, the exponential limit law for return resp. hitting times, taking the limit along the full sequence of cylinders, have been obtained only in positive-entropy processes satisfying some strong mixing conditions of Rossenblatt type.

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Extremes and Recurrence in Dynamical Systems

Lucarini¹,

Faranda²,

Freitas³

et al. 2016

194

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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Asymptotics for Return Times of Rank-One Systems

Cited by 8 publications

References 13 publications

Matrix Technique Representation for Second Return Time Asymptotics

Matrix Technique Representation for Second Return Time Asymptotics

Exponential return times in a zero-entropy process

Extremes and Recurrence in Dynamical Systems

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