Shortest distance between observed orbits in distinct Dynamical Systems

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 $(X, T, μ)$ and $(X, S, η)$ 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 $\widehat{m}<em>n<sup>f(x,y)$ is governed by the correlation dimensions of the pushforward measures $f</em><em>μ$ and $f_</em>η$. Under suitable mixing assumptions, we show that the asymptotic rate of decay of $\widehat{m}<em>n<sup>f(x,y)$ is governed by the symmetric Rényi divergence of the pushforward measures $f</em><em>μ$ and $f_</em>η$. 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.