## Asymptotic pairs in positive-entropy systems

*Ergod. Th. & Dynam. Syst.*, **22**, 671-686, 2002.

### 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* is not equal to *y* and .
More precisely we consider a
*T*-ergodic measure
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
-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.

Paper:
[arXiv:1901.00327]
[pdf]