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

TL;DR

This paper introduces a new condition for sequences of integers and demonstrates that positive topological entropy guarantees the existence of special mixing subsets and chaos phenomena along these sequences.

Contribution

It establishes a novel condition for sequences and links positive entropy to the existence of $\Delta$-weakly mixing subsets and Li-Yorke chaos along polynomial times.

Findings

Positive topological entropy implies $\Delta$-weakly mixing subsets.

Existence of $\Delta$-weakly mixing subsets along sequences satisfying Condition $(**)$.

Positive entropy leads to multi-variant Li-Yorke chaos along polynomial times.

Abstract

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 $\sigma$-algebra is a characteristic $\sigma$-algebra for the averages along a collection satisfying Condition $(**)$. 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.