Rare events for the Manneville–Pomeau map

@article{Freitas2015RareEF,
  title={Rare events for the Manneville–Pomeau map},
  author={Ana Cristina Moreira Freitas and Jorge Milhazes Freitas and Mike Todd and Sandro Vaienti},
  journal={Stochastic Processes and their Applications},
  year={2015},
  volume={126},
  pages={3463-3479}
}
View PDF on arXiv
28 Citations
Background Citations
6
Methods Citations
5

28 Citations

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

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…

Dichotomy results for eventually always hitting time statistics and almost sure growth of extremes

Suppose (f,X , μ) is a measure preserving dynamical system and φ : X → R a measurable function. Consider the maximum process Mn := max{X1, . . . , Xn}, where Xi = φ ◦ f i−1 is a time series of…

EXTREMAL DICHOTOMY FOR TORAL HYPERBOLIC AUTOMORPHISMS

Abstract. We consider the extreme value theory of a hyperbolic toral automorphism T : T2 ! T2 showing that, if a Hölder observation is a function of a Euclidean-type distance to a non-periodic point…

Extremal dichotomy for uniformly hyperbolic systems

We consider the extreme value theory of a hyperbolic toral automorphism showing that, if a Hölder observation φ is a function of a Euclidean-type distance to a non-periodic point ζ and is strictly…

Extreme Value Laws for Dynamical Systems with Countable Extremal Sets

We consider stationary stochastic processes arising from dynamical systems by evaluating a given observable along the orbits of the system. We focus on the extremal behaviour of the process, which is…

Convergence of marked point processes of excesses for dynamical systems

We consider stochastic processes arising from dynamical systems simply by evaluating an observable function along the orbits of the system and study marked point processes associated to extremal…

Hitting Times and Positions in Rare Events

We establish abstract limit theorems which provide sufficient conditions for a sequence $(A_{l})$ of rare events in an ergodic probability preserving dynamical system to exhibit Poisson asymptotics,…

Clustering of extreme events created by multiple correlated maxima

Entry and return times for semi-flows

Haydn, Lacroix and Vaienti (2005 Ann. Probab. 33 2043–50) proved that, for a given ergodic map, the entry time distribution converges in the small target limit, if and only if the corresponding…

Hitting Times and Positions in Rare Events

— We establish abstract limit theorems which provide sufficient conditions for a sequence (Al) of rare events in an ergodic probability preserving dynamical system to exhibit Poisson asymptotics, and…

References

SHOWING 1-10 OF 45 REFERENCES

Convergence of rare event point processes to the Poisson process for planar billiards

We show that for planar dispersing billiards the distribution of return times is, in the limit, Poisson for metric balls almost everywhere w.r.t. the SRB (Sinai–Ruelle–Bowen) measure. Since the…

Extreme values for Benedicks–Carleson quadratic maps

Abstract We consider the quadratic family of maps given by fa(x)=1−ax2 with x∈[−1,1], where a is a Benedicks–Carleson parameter. For each of these chaotic dynamical systems we study the extreme value…

Poisson and compound Poisson approximations in conventional and nonconventional setups

It was shown in Kifer (Israel J Math, 2013) that for any subshift of finite type considered with a Gibbs invariant measure the numbers of multiple recurrencies to shrinking cylindrical neighborhoods…

The extremal index, hitting time statistics and periodicity

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

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of certain non-uniformly hyperbolic dynamical systems is compound Poisson. The…

Poisson approximation for the number of visits to balls in non-uniformly hyperbolic dynamical systems

Abstract We study the number of visits to balls Br(x), up to time t/μ(Br(x)), for a class of non-uniformly hyperbolic dynamical systems, where μ is the Sinai–Ruelle–Bowen measure. Outside a set of…

Hitting and returning to rare events for all alpha-mixing processes

A Poisson limit theorem for toral automorphisms

We introduce a new method of proving Poisson limit laws in the theory of dynamical systems, which is based on the Chen-Stein method (7, 20]) combined with the analysis of the homoclinic Laplace…

Statistical properties of long return times in type I intermittency

We study a class of maps of the unit interval with a neutral fixed point such äs those modelling Pomeau-Manneville type l intermittency. We construct the invariant ergodic probability measure…

Hitting time statistics and extreme value theory

We consider discrete time dynamical systems and show the link between Hitting Time Statistics (the distribution of the first time points land in asymptotically small sets) and Extreme Value Theory…