Three Zutot

TL;DR

This paper explores advanced topics in dynamical systems, solving open problems related to Furstenberg entropy and rigid actions, and presents a novel example illustrating unexpected properties of ergodic actions in non-amenable groups.

Contribution

It addresses open problems in Furstenberg entropy and rigid actions, and introduces a new example revealing surprising features of ergodic actions for non-amenable groups.

Findings

Solved open problems on Furstenberg entropy and rigid actions

Provided a new example of ergodic actions with unexpected properties

Enhanced understanding of dynamical systems in non-amenable groups

Abstract

Three topics in dynamical systems are discussed. In the first two sections we solve some open problems concerning, respectively, Furstenberg entropy of stationary dynamical systems, and uniformly rigid actions admitting a weakly mixing fully supported invariant probability measure. In the third section we provide a new example that displays some unexpected properties of strictly ergodic actions of non-amenable groups.