On the shortest distance between orbits and the longest common substring problem

TL;DR

This paper investigates the shortest distance between orbits in dynamical systems and relates it to the longest common substring problem, revealing how decay rates depend on mixing properties and irrational rotation parameters.

Contribution

It extends existing results on sequence matching to $eta$-mixing processes and analyzes the decay of shortest distances in dynamical systems under various conditions.

Findings

Decay of shortest distance depends on correlation dimension under mixing conditions.

Different behavior observed for irrational rotations based on irrational exponent.

Extended Arratia and Waterman's results to $eta$-mixing processes with exponential decay.

Abstract

In this paper, we study the behaviour of the shortest distance between orbits and show that under some rapidly mixing conditions, the decay of the shortest distance depends on the correlation dimension. For irrational rotations, we prove a different behaviour depending on the irrational exponent of the angle of the rotation. For random processes, this problem corresponds to the longest common substring problem. We extend the result of Arratia and Waterman on sequence matching to $\alpha$-mixing processes with exponential decay.