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.
Categories: math.DS