Almost sure orbits closeness

Abstract

We consider the minimal distance between orbits of measure preserving dynamical systems. In the spirit of dynamical shrinking target problems we identify distance rates for which almost sure asymptotic closeness properties can be ensured. More precisely, we consider the set $E_n$ of pairs of points whose orbits up to time $n$ have minimal distance to each other less than the threshold $r_n$. We obtain bounds on the sequence $(r_n)_n$ to guarantee that $\limsup_{n}E_n$ and $\liminf_{n} E_n$ are sets of measure 0 or 1. Results for the measure 0 case are obtained in broad generality while the measure one case requires assumptions of exponential mixing for at least one of the systems. We also consider the analogous question of the minimal distance of points within a single orbit of one dimensional exponentially mixing dynamical systems.