W-measurable sensitivity of semigroup actions
Volume 163 / 2021
Abstract
This paper studies the notion of W-measurable sensitivity in the context of countable 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 nonsingular dynamical systems which states that 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 showing that W-measurable sensitivity is not preserved under factors. We also consider the restriction of W-measurably sensitive semigroup actions to subsemigroups and show that the restriction remains W-measurably sensitive when the subsemigroup is large enough (e.g. when the subsemigroups are syndetic or thick).
