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Limiting Entry and Return Times Distribution for Arbitrary Null Sets
Communications in Mathematical Physics ( IF 2.4 ) Pub Date : 2020-07-06 , DOI: 10.1007/s00220-020-03795-0
Nicolai Haydn , Sandro Vaienti

We describe an approach that allows us to deduce the limiting return times distribution for arbitrary sets to be compound Poisson distributed. We establish a relation between the limiting return times distribution and the probability of the cluster sizes, where clusters consist of the portion of points that have finite return times in the limit where random return times go to infinity. In the special case of periodic points we recover the known Polya-Aeppli distribution which is associated with geometrically distributed cluster sizes. We apply this method to several examples the most important of which is synchronisation of coupled map lattices. For the invariant absolutely continuous measure we establish that the returns to the diagonal is compound Poisson distributed where the coefficients are given by certain integrals along the diagonal.

中文翻译:

限制任意空集的进入和返回时间分布

我们描述了一种方法,它允许我们将任意集合的极限返回时间分布推导出为复合泊松分布。我们建立了极限返回时间分布和簇大小概率之间的关系,其中簇由在随机返回时间趋于无穷大的极限内具有有限返回时间的点的部分组成。在周期点的特殊情况下,我们恢复了已知的 Polya-Aeppli 分布,它与几何分布的簇大小相关联。我们将此方法应用于几个示例,其中最重要的是耦合地图格子的同步。对于不变的绝对连续度量,我们确定对角线的收益是复合泊松分布,其中系数由沿对角线的某些积分给出。
更新日期:2020-07-06
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