# Asymptotic pairs in positive-entropy systems

## TL;DR

This paper proves that positive-entropy dynamical systems contain abundant asymptotic pairs, with measure-theoretic and topological implications, including the density of such pairs among entropy pairs.

## Contribution

It establishes the existence and measure-theoretic prevalence of asymptotic pairs in positive-entropy systems, and explores their properties under invertibility.

## Key findings

- Proper asymptotic pairs exist in positive-entropy systems.
- Almost every point belongs to a proper asymptotic pair under an ergodic measure.
- Asymptotic pairs are dense among topological entropy pairs.

## 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 $1$. 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.