Asymptotic pairs in positive-entropy systems

@article{Blanchard2002AsymptoticPI,
  title={Asymptotic pairs in positive-entropy systems},
  author={Franccois Blanchard and Bernard Host and Sylvie Ruette},
  journal={Ergodic Theory and Dynamical Systems},
  year={2002},
  volume={22},
  pages={671 - 686}
}
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… 
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