On recurrence over subsets and weak mixing

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.U A; V/ 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 nC 1).