Asymptotic pairs in positive-entropy systems

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 equal y and lim(n)-->+infinity 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.