$\Delta$-weakly mixing subsets along a collection of sequences of integers

Abstract

In this paper, we propose a mild condition, named Condition $(\ast \ast )$, for collections of sequence of integers and show that for any measure preserving system the Pinsker $\sigma$-algebra is a characteristic $\sigma$-algebra for the averages along a collection satisfying Condition $(\ast \ast )$. We introduce the notion of $\Delta$-weakly mixing subsets along a collection of sequences of integers and show that positive topological entropy implies the existence of $\Delta$-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.

References in corpus (7)

• Recent development of chaos theory in topological dynamics
• Stable sets and mean Li-Yorke chaos in positive entropy systems
• Some open problems on multiple ergodic averages
• Positive topological entropy and $\Delta$-weakly mixing sets
• Mean Li-Yorke chaos along some good sequences
• Positive entropy implies chaos along any infinite sequence
• Localization of mixing property via Furstenberg families