Rare Events for Cantor Target Sets

Freitas, Ana Cristina Moreira; Freitas, Jorge Milhazes; Rodrigues, Fagner B.; Soares, Jorge Valentim

doi:10.1007/s00220-020-03794-1

Cited by 10 publications

(37 citation statements)

References 40 publications

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“…Then we use the conformality of the factor measure µ along the x-axis and we get q 0 = 1/2, giving also an extremal index equal to 0.5. It is interesting to compare this result with that in [32] described in section 7.1 and with a global approach to the Cantor set as a rare event: the EV found there was 1/3.…”

Section: 2mentioning

confidence: 67%

“…The first study dealing with the ternary Cantor set in connection with EVT was mostly numerical and it was given in [53]: the authors conjectured the existence of a limiting extreme value law with an EI equal to 1. A rigorous proof appeared recently in the paper [32]; in particular, the authors introduced the observable…”

Section: Open Systemsmentioning

confidence: 99%

“…where C n is the disjoint union of the 2 n sets in the construction of the Cantor set C. 11 Notice that the function φ will have his maximum (infinity) on the Cantor set, otherwise it says how fare we are from it: it is called the Cantor ladder function in [53,32]. The probability was chosen as the Lebesgue measure Leb on the unit interval.…”

Section: Open Systemsmentioning

confidence: 99%

“…-if we choose z as a periodic point (they are dense in C), of minimal period p, we get for the EI θ : [32] and our local approaches to the EVT distribution just described, considered the Cantor set as the non-wandering set of a dynamical system defined on the unit interval. One could consider the dynamics defined directly on the Cantor sets, which means to study the system (C, T |C ).…”

Section: Open Systemsmentioning

confidence: 99%

See 3 more Smart Citations

Extreme value distributions of observation recurrences

Caby¹,

Faranda²,

Vaienti³

et al. 2020

Nonlinearity

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show abstract

Section: 2mentioning

confidence: 67%

Section: Open Systemsmentioning

confidence: 99%

Section: Open Systemsmentioning

confidence: 99%

Section: Open Systemsmentioning

confidence: 99%

See 2 more Smart Citations

Extreme value distributions of observation recurrences

Caby¹,

Faranda²,

Vaienti³

et al. 2020

Nonlinearity

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show abstract

“…We take, as in the previous example, the following observable: 12) for some δ > 0 and g as described above. In this case, we also have a counterexample where the EI cannot be identified as the reciprocal of the mean limiting cluster size distribution.…”

Section: Dynamical Counterexample With Periodic Behaviourmentioning

confidence: 99%

Dynamical counterexamples regarding the extremal index and the mean of the limiting cluster size distribution

Abadi

Freitas

2020

J. London Math. Soc.

Self Cite

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The extremal index (EI) is a parameter that measures the intensity of clustering of rare events and is usually equal to the reciprocal of the mean of the limiting cluster size distribution. We show how to build dynamically generated stochastic processes with an EI for which that equality does not hold. The mechanism used to build such counterexamples is based on considering observable functions maximised at least two points of the phase space, where one of them is an indifferent periodic point and another one is either a repelling periodic point or a non-periodic point. The occurrence of extreme events is then tied to the entrance and recurrence to the vicinities of those points. This enables to mix the behaviour of an EI equal to 0 with that of an EI larger than 0. Using bi-dimensional point processes, we explain how mass escapes in order to destroy the usual relation. We also perform a study about the formulae to compute the cluster size distribution introduced earlier and prove that ergodicity is enough to establish that the finite versions of the reciprocal of the EI and of the mean of the cluster size distribution do coincide. Contents

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On the Computation of the Extremal Index for Time Series

et al. 2019

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The extremal index is a quantity introduced in extreme value theory to measure the presence of clusters of exceedances. In the dynamical systems framework, it provides important information about the dynamics of the underlying systems. In this paper we provide a review of the meaning of the extremal index in dynamical systems. Depending on the observables used, this quantity can inform on local properties of attractors such as periodicity, stability and persistence in phase space, or on global properties such as the Lyapunov exponents. We also introduce a new estimator of the extremal index and shows its relation with those previously introduced in the statistical literature. We reserve a particular focus to the systems perturbed with noise as they are a good paradigm of many natural phenomena. Different kind of noises are investigated in the annealed and quenched situations. Applications to climate data are also presented.

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Rare Events for Cantor Target Sets

Cited by 10 publications

References 40 publications

Extreme value distributions of observation recurrences

Extreme value distributions of observation recurrences

Dynamical counterexamples regarding the extremal index and the mean of the limiting cluster size distribution

On the Computation of the Extremal Index for Time Series

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