Minimal distance between random orbits

TL;DR

This paper investigates the minimal distance between orbit segments in random dynamical systems, revealing complex asymptotic behavior involving two correlation dimensions, contrasting with known results in non-random systems.

Contribution

It extends the understanding of orbit distances to quenched random systems, showing a more intricate asymptotic behavior involving two correlation dimensions.

Findings

Asymptotic behavior involves two correlation dimensions.

Non-smooth asymptotic exponent due to quenched randomness.

Contrasts with simpler annealed or non-random cases.

Abstract

We study the minimal distance between two orbit segments of length n, in a random dynamical system with sufficiently good mixing properties. This problem has already been solved in non-random dynamical system, and on average in random dynamical systems (the so-called annealed version of the problem): it is known that the asymptotic behavior for this question is given by a dimension-like quantity associated to the invariant measure, called its correlation dimension (or R{\'e}nyi entropy). We study the analogous quenched question, and show that the asymptotic behavior is more involved: two correlation dimensions show up, giving rise to a non-smooth behavior of the associated asymptotic exponent.