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Spatio-temporal Poisson processes for visits to small sets
Israel Journal of Mathematics ( IF 0.8 ) Pub Date : 2020-10-01 , DOI: 10.1007/s11856-020-2074-0
Françoise Pène , Benoît Saussol

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.

中文翻译:

访问小集合的时空泊松过程

对于许多保留度量的动态系统 $(\Omega,T,m)$,连续击中小集合的时间可以通过实线上的泊松过程很好地近似。在这项工作中,我们定义了一个新过程,该过程不仅记录了对集合 $A$ 的连续访问次数 $n$,而且还记录了轨道 $A$ 中的位置 $T^n(x)$,在限制 $m(A)\to0$。我们获得了这个过程的收敛,适当地归一化,在一些去相关条件下,在时间和空间上泊松点过程。我们介绍了双曲线图和 SRB 测量的几个新应用,包括周期点邻域的情况,以及一些台球,如西奈台球、布尼莫维奇体育场和钻石台球。
更新日期:2020-10-01
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