Spatio-temporal Poisson processes for visits to small sets

Françoise Pène; Benoit Saussol

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arXiv · 1803.06865

Spatio-temporal Poisson processes for visits to small sets

Françoise Pène·Benoit Saussol

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Abstract

For many measure preserving dynamical systems $(\Omega,T,m)$ the successive hitting times to a small set is well approximated by a Poisson process on the real line. In this work we define a new process obtained from recording not only the successive times $n$ of visits to a set $A$, but also the position $T^n(x)$ in $A$ of the orbit, in the limit where $m(A)\to0$. We obtain a convergence of this process, suitably normalized, to a Poisson point process in time and space under some decorrelation condition. We present several new applications to hyperbolic maps and SRB measures, including the case of a neighborhood of a periodic point, and some billiards such as Sinai billiards, Bunimovich stadium and diamond billiard.

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BibTeX RIS

Françoise Pène, Benoit Saussol. 2018-03-19. Spatio-temporal Poisson processes for visits to small sets. https://arxiv.org/abs/1803.06865

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