Mixing and proximal cells along sequences

Abstract

A dynamical system (X, T) is -transitive if for each pair of open and non-empty subsets U and V of X, , where is a collection of subsets of that is hereditary upward. (X, T) is -mixing if (X × X, T × T) is -transitive. For a subset S of , (x, y) ϵ X × X is S-proximal if and the S-proximal cell PS(x) is the set of points that are S-proximal to x ϵ X. We show that if (X, T) is -mixing, then for each (the dual family of ) and x ϵ X, PS(x) is a dense Gδ subset of X, and when (X, T) is minimal and is a filter the reciprocal is true. Moreover, other conditions under which the reciprocal is true are obtained. Finally the structure of proximal cells for -mixing systems is discussed, and a new and simpler proof of the Xiong–Yang theorem is presented.