Δ-weak mixing and asynchronous Li-Yorke chaos

In this paper, we investigate Δ-weakly mixing sets and asynchronous Li–Yorke chaos. First, we prove that in a measure-theoretic sense, every ergodic, non-distal measure preserving system contains many Δ-weakly mixing subsets. Second, we show that every non-trivial transitive system contains an uncountable ( r , s ) -scrambled set for each pair of distinct r , s ∈ N ; under the stronger assumption of total transitivity, there exists a Mycielski subset that is ( r , s ) -scrambled simultaneously for all distinct r , s ∈ N .