An explicit scattering, non-weakly mixing example and weak disjointness

Abstract

By a dynamical system we mean a pair (X,T), whereX is a compact metric space and T:X→X is surjective and continuous.We study weak disjointness in topological dynamics.(X,T) is scattering iff it is weakly disjoint from all minimal systems and(X,T) is strongly scattering iff it is weakly disjoint from all E-systems,i.e. transitive systems having invariant measures with full support.It is clear that a weakly mixing system is strongly scattering and the latter is scattering.An existential proof of scattering and a non-weakly mixing example is obtainedby Akin and Glasner (2001 J. Anal. Math. 84 243-86). In this paper, we will give an explicit examplewhich is strongly scattering and not weakly mixing. We also define extremescattering, weak scattering and study the relationships of the various definitions.For a dynamical property P stronger than transitivity, let P⋏ be theproperty such that a system has P⋏ iff it is weakly disjointfrom any system having P. We show that P⋏ = P⋏⋏⋏. Moreover,we prove that (thickly syndetic-transitive)⋏ = piecewise-syndetic-transitive and (piecewise-syndetic-transitive)⋏ = thickly syndetic-transitive.