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Asymptotic pairs in positive-entropy systems

Published online by Cambridge University Press:  19 June 2002


F. BLANCHARD
Affiliation:
Institut de Mathématiques de Luminy, CNRS UPR 9016, 163, avenue de Luminy, case 907, 13288 Marseille Cedex 9, France (e-mail: {blanchar,ruette}@iml.univ-mrs.fr)
B. HOST
Affiliation:
Université de Marne la Vallée, Cité Descartes, 5, boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée Cedex 2, France (e-mail: host@math.univ-mlv.fr)
S. RUETTE
Affiliation:
Institut de Mathématiques de Luminy, CNRS UPR 9016, 163, avenue de Luminy, case 907, 13288 Marseille Cedex 9, France (e-mail: {blanchar,ruette}@iml.univ-mrs.fr)

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.


Type
Research Article
Copyright
2002 Cambridge University Press

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