On recurrence over subsets and weak mixing

Jian Li, Piotr Oprocha and Guohua Zhang

Vol. 277 (2015), No. 2, 399–424

DOI: 10.2140/pjm.2015.277.399

Abstract

We study properties of weakly mixing sets (of order $n$) in relation to proximality, sensitivity, scrambled tuples, Xiong chaotic sets and independent sets. Our main emphasis is on the structure of the set of transfer times $N\left(U\cap A,V\right)$ between open sets $U$ and $V$, both intersecting a weakly mixing set $A$. We find several conditions on properties of the set $A$ that are equivalent to weak mixing.

We also prove that on topological graphs weakly mixing sets of order $2$ can be approximated arbitrarily closely by a weakly mixing set of all orders. This property is known to hold on the unit interval but is not true in general (there are systems with weakly mixing sets of order $n$ but not $n+1$).

Keywords

weakly mixing sets, Xiong chaotic sets, proximality, scrambled tuples, independent sets, topological graphs

Mathematical Subject Classification 2010

Primary: 37B05

Secondary: 37B40, 37E05, 37E25

Milestones

Received: 27 November 2013

Revised: 10 December 2014

Accepted: 10 March 2015

Published: 15 September 2015