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 $x\ne y$ 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 $1$. 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 with $x$ 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.

Published in 2002