We consider rapidly mixing dynamical systems and link the decay of the shortest distance between multiple orbits with the generalized fractal dimension. We apply this result to multidimensional expanding maps and extend it to the realm of random dynamical systems. For random sequences, we obtain a relation between the longest common substring between multiple sequences and the generalized Rényi entropy. Applications to Markov chains and Gibbs states are given.

中文翻译：

---

多轨道和广义分形维数之间的最短距离

我们考虑快速混合动力系统，并将多个轨道之间最短距离的衰减与广义分形维数联系起来。我们将此结果应用于多维扩展图，并将其扩展到随机动力学系统的领域。对于随机序列，我们获得多个序列之间最长的公共子串与广义Rényi熵之间的关系。给出了对马尔可夫链和吉布斯状态的应用。