Asymptotic pairs in positive-entropy systems
By François Blanchard, B. Host, Sylvie Ruette. Ergodic Theory and Dynamical Systems, 2002
88 citations recorded by the source corpus.
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.
View Paper: https://www.semanticscholar.org/paper/54ad54dbffc1f0398cba140348a8bd5f69901cf0
How researchers see this paper
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- Jian Li
- Xiangdong Ye
- Eli Glasner
- Lai-Sang Young
- François Blanchard
- Wen Huang
- Tomasz Downarowicz
- Hanfeng Li
Citations in the corpus
- Entropy in dynamical systems (2003, methodology): “There is a quite complicated example in [Blanchard et al., 2002].”
- Topological chaos: what may this mean? (2008, background): “Contrarily completely scrambled systems have been proved to have entropy zero [12], which again suggests that the property is not a strongly chaotic one.”
- Homoclinic groups, IE groups, and expansive algebraic actions (2011, unknown): “[10].”
- On tame dynamical systems (2006, methodology): “1.11. Remark. In the proof of Theorem 1.10, with slight modiflcations, one can use instead of the results in [5] a theorem of Blanchard, Host and Ruette [ 6 ] on the abundance of asymptotic pairs is a system (X;T) with positive topological entropy.”
- Recent development of chaos theory in topological dynamics (2015, background): “Note that in [16] Blanchard et al.”
- Stable sets and mean Li–Yorke chaos in positive entropy systems (2012, methodology): “We remark that the partition P constructed in Lemma 2.2 is an excellent partition (see [6] or [26]), which is a key tool in the proof of our main results.”
- Local entropy theory (2009, methodology): “This was done by Blanchard et al in [9].”