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：

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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。