Shortest distance between multiple orbits and generalized fractal dimensions

TL;DR

This paper explores the relationship between the shortest distance among multiple orbits in dynamical systems and their fractal dimensions, extending results to random systems and applications like Markov chains.

Contribution

It introduces a novel connection between orbit distances and generalized fractal dimensions, extending the theory to random dynamical systems and sequences.

Findings

Decay of orbit distances linked to fractal dimensions

Relation between longest common substrings and Rènyi entropy

Applications to Markov chains and Gibbs states

Abstract

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 R\'enyi entropy. Applications to Markov chains, Gibbs states and the stochastic scrabble are given.