标题： 典型点的迭代之间的最近距离 显示英文标题

标题： Closest Distance between Iterates of Typical Points

摘要： 典型点的前$n$次迭代之间的最短距离可以通过某些具有Gibbs测度的动力系统的一个对数规则来量化。 我们在两种情况下证明了这一点。 对于具有至多可数无限字母且相对于局部Hölder势函数具有Gibbs测度的拓扑混合马尔可夫移位，我们证明了典型点的最长公共子串的渐近长度收敛，且极限取决于Rényi熵。 对于具有Gibbs-Markov结构的区间映射，我们证明了一个类似的规则，将Gibbs测度的相关维数与由典型点生成的轨道中两个迭代之间的最短距离联系起来。 显示英文摘要

摘要： The shortest distance between the first $n$ iterates of a typical point can be quantified with a log rule for some dynamical systems admitting Gibbs measures. We show this in two settings. For topologically mixing Markov shifts with at most countably infinite alphabet admitting a Gibbs measure with respect to a locally H\"{o}lder potential, we prove the asymptotic length of the longest common substring for a typical point converges and the limit depends on the R\'{e}nyi entropy. For interval maps with the Gibbs-Markov structure, we prove a similar rule relating the correlation dimension of Gibbs measures with the shortest distance between two iterates in the orbit generated by a typical point.