The shortest distance between orbits of two different dynamical systems

Abstract

We study the shortest distance, denoted by \(\widehat{m}_{n}(x,y)\), between distinct n-step orbits of different dynamical systems. The main results concern the asymptotic behavior of \(\widehat{m}_{n}(x,y)\) as n increases. We prove that the asymptotic behavior of \(\widehat{m}_{n}(x,y)\) as n increases is characterized by the correlation dimension of the invariant measures of the systems, provided both systems exhibit exponential mixing in a suitable functional analytic framework. Applications of the results are provided for rotations and expanding maps.