Shortest distance between observed orbits in distinct Dynamical Systems

TL;DR

This paper studies how the shortest observed orbit distances between two different dynamical systems decay over time, linking this behavior to the measures' divergence and extending previous single-system analyses.

Contribution

It introduces a framework for analyzing the asymptotic decay of shortest orbit distances between two distinct systems using measure divergence, generalizing prior single-system results.

Findings

Decay rate governed by correlation dimensions or Rényi divergence

Results extend to random dynamical systems

Applicable to systems with mixing assumptions

Abstract

In this paper, we investigate the asymptotic behavior of the shortest distance between observed orbits in two distinct dynamical systems. Given two measure-preserving transformations $(X, T, \mu)$ and $(X, S, \eta)$ and a Lipschitz observation function $f$, we define \[ \widehat{m}_n^f(x,y) = \min_{i=0,\ldots,n-1} d\big(f(T^i x), f(S^i y)\big). \] %Under suitable mixing assumptions, we show that the asymptotic rate of decay of $\widehat{m}_n^f(x,y)$ is governed by the correlation dimensions of the pushforward measures $f_*\mu$ and $f_*\eta$. Under suitable mixing assumptions, we show that the asymptotic rate of decay of $\widehat{m}_n^f(x,y)$ is governed by the symmetric R\'enyi divergence of the pushforward measures $f_*\mu$ and $f_*\eta$. Our results generalize previous work that consider either a single system or the unobserved case. In addition, we discuss the extension of these results to random dynamical systems and illustrate the applicability of the approach with an example.