Asymptotic pairs in positive-entropy systems

Abstract

We show that in a topological dynamical system (X,T) of positive entropy there exist proper (positively) asymptotic pairs, that is, pairs (x,y) such that x\not= y and \lim_{n\to +\infty} d(T^n x,T^n y)=0. More precisely we consider a T-ergodic measure \mu of positive entropy and prove that the set of points that belong to a proper asymptotic pair is of measure one. When T is invertible, the stable classes (i.e. the equivalence classes for the asymptotic equivalence) are not stable under T^{-1}: for \mu-almost every x there are uncountably many y that are asymptotic to x and such that (x,y) is a Li–Yorke pair with respect to T^{-1}. We also show that asymptotic pairs are dense in the set of topological entropy pairs.