The chaotic behavior of a mixing measure-preserving transformation (Q1586662)

Given a Borel probability space \((X, {\mathcal B}(X), \mu)\) where \(X\) is a second countable space and a measure preserving transformation \(f\) on \(X\), the existence of finitely many chaotic sets \(C\) (of \(f\) or \(f^n\) with respect to a given sequence of \((p_i)\) of positive integers) satisfying \(C\cap D\not = \emptyset\) whenever \(\mu (D) >0\) is proved if the system fulfills one of the following conditions: (i) \(f\) is mixing on \((p_i)\); (ii) \(X\) is compact and \(f\) is weakly mixing; (iii) \(f\) is strongly mixing.