Paper detail

$Δ$-weakly mixing subsets along a collection of sequences of integers

In this paper, we propose a mild condition, named Condition $(**)$, for collections of sequence of integers and show that for any measure preserving system the Pinsker $σ$-algebra is a characteristic $σ$-algebra for the averages along a collection satisfying Condition $(**)$. We introduce the notion of $Δ$-weakly mixing subsets along a collection of sequences of integers and show that positive topological entropy implies the existence of $Δ$-weakly mixing subsets along a collection of "good" sequences. As a consequence, we show that positive topological entropy implies multi-variant Li-Yorke chaos along polynomial times of the shift prime numbers.