Orbits closeness for slowly mixing dynamical systems

TL;DR

This paper investigates how the shortest distance between two orbits in slowly mixing dynamical systems scales with orbit length, providing new bounds, extending results to flows, and presenting an example with no scaling limit.

Contribution

It proves that the shortest orbit distance scales polynomially in slowly mixing systems, extends these results to flows, and offers a counterexample with no scaling limit.

Findings

Shortest orbit distance scales polynomially with orbit length.

Results extend to continuous flows.

Counterexample shows no universal scaling limit.

Abstract

Given a dynamical system, we prove that the shortest distance between two $n$-orbits scales like $n$ to a power even when the system has slow mixing properties, thus building and improving on results of Barros, Liao and the first author. We also extend these results to flows. Finally, we give an example for which the shortest distance between two orbits has no scaling limit.