Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems | Zendy

Nicolai Haydn | Zendy; Kasia Wasilewska | Zendy

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Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems

Author(s) -

Nicolai Haydn,

Kasia Wasilewska

Publication year - 2015

Publication title -

discrete and continuous dynamical systems

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.289

H-Index - 70

eISSN - 1553-5231

pISSN - 1078-0947

DOI - 10.3934/dcds.2016.36.2585

Subject(s) - mathematics , attractor , logarithm , invariant measure , rate of convergence , uniform limit theorem , poisson distribution , dynamical systems theory , limit set , invariant (physics) , mathematical analysis , limit (mathematics) , limiting , statistics , physics , computer science , mechanical engineering , quantum mechanics , engineering , computer network , channel (broadcasting) , mathematical physics , ergodic theory

We show that for systems that allow a Young tower construction with polynomially decaying correlations the return times to metric balls are in the limit Poisson distributed. We also provide error terms which are powers of logarithm of the radius. In order to get those uniform rates of convergence the balls centres have to avoid a set whose size is estimated to be of similar order. This result can be applied to non-uniformly hyperbolic maps and to any invariant measure that satisfies a weak regularity condition. In particular it shows that the return times to balls is Poissonian for SRB measures on attractors.

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