F-mixing and weak disjointness

A topological system (X,f) is F-transitive if for each pair of opene subsets U and V of X, Nf(U,V)={n∈Z+: fnU∩V≠∅}∈F, where F is a collection of subsets of Z+ which is hereditary upward. (X,f) is F-mixing if (X×X,f×f) is F-transitive. In this paper F-mixing systems are characterized in terms of the chaoticity of the systems. Moreover, weak disjointness is studied via family. We will give conditions such that a dual theorem of the Weiss–Akin–Glasner theorem holds. Examples with this dual theorem fails for some “good” families are obtained.