Shortest distance between observed orbits in distinct Dynamical Systems

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.