Family-independence for topological and measurable dynamics

TL;DR

This paper explores various notions of independence in topological and measurable dynamics, establishing key equivalences and non-existence results, and constructing examples to deepen understanding of dynamical system properties.

Contribution

It introduces the concept of F-independence in both settings, proves non-existence of certain independent systems, and links independence with entropy and mixing properties.

Findings

No non-trivial syndetic-independent measurable systems.

Positive-density independence corresponds to completely positive entropy.

Weakly mixing systems are characterized by IP-independence.

Abstract

For a family F (a collection of subsets of Z_+), the notion of F-independence is defined both for topological dynamics (t.d.s.) and measurable dynamics (m.d.s.). It is shown that there is no non-trivial {syndetic}-independent m.d.s.; a m.d.s. is {positive-density}-independent if and only if it has completely positive entropy; and a m.d.s. is weakly mixing if and only if it is {IP}-independent. For a t.d.s. it is proved that there is no non-trivial minimal {syndetic}-independent system; a t.d.s. is weakly mixing if and only if it is {IP}-independent. Moreover, a non-trivial proximal topological K system is constructed, and a topological proof of the fact that minimal topological K implies strong mixing is presented.