W-measurable sensitivity of semigroup actions

TL;DR

This paper extends the concept of W-measurable sensitivity from ergodic dynamical systems to semigroup actions, providing classification results, counterexamples, and conditions under which sensitivity is preserved.

Contribution

It generalizes the classification of W-measurable sensitivity to semigroup actions and analyzes its preservation under factors and sub-semigroup restrictions.

Findings

W-measurable sensitivity is not preserved under factors.

Sensitivity is preserved when restricting to large enough sub-semigroups.

Classification of semigroup actions into sensitive or isometric types.

Abstract

This paper studies the notion of W-measurable sensitivity in the context of semigroup actions. W-measurable sensitivity is a measurable generalization of sensitive dependence on initial conditions. In 2012, Grigoriev et. al. proved a classification result of conservative ergodic dynamical systems that states all are either W-measurably sensitive or act by isometries with respect to some metric and have refined structure. We generalize this result to a class of semigroup actions. Furthermore, a counterexample is provided that shows W-measurable sensitivity is not preserved under factors. We also consider the restriction of W-measurably sensitive semigroup actions to sub-semigroups and show that the restriction remains W-measurably sensitive when the sub-semigroup is large enough (e.g. when the sub-semigroups are syndetic or thick).