Positive topological entropy implies chaos DC2

Abstract:

Using methods of entropy in ergodic theory, we prove that positive topological entropy implies chaos DC2. That is, if a system $(X,T)$ has positive topological entropy, then there exists an uncountable set $E$ such that for any two distinct points $x,y$ in $E$, \[ \liminf _{n\to \infty } \frac 1n \sum _{i=1}^n \mathsf {dist}(T^ix,T^iy)=0 \ \ \ \text {and} \ \ \limsup _{n\to \infty } \frac 1n \sum _{i=1}^n \mathsf {dist}(T^ix,T^iy)>0 . \]