Topological size of scrambled sets

Abstract

A subset $S$ of a topological dynamical system $(X,f)$ containing at least two points is called a <i>scrambled set</i> if for any $x,y\in S$ with $x\neq y$ one has $$\liminf_{n\to \infty} d(f^n(x), f^n(y)) = 0 \quad \hbox{and} \quad \limsup_{n\to \infty}