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W-measurable sensitivity of semigroup actions

Volume 163 / 2021

Francisc Bozgan, Anthony Sanchez, Cesar E. Silva, Jack Spielberg, David Stevens, Jane Wang Colloquium Mathematicum 163 (2021), 113-129 MSC: Primary 37A40; Secondary 37A05, 20M99. DOI: 10.4064/cm7815-2-2020 Published online: 15 June 2020


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).


  • Francisc BozganDepartment of Mathematics
    University of Chicago
    Chicago, IL 60637, U.S.A.
  • Anthony SanchezDepartment of Mathematics
    University of Washington
    Box 354350
    Seattle, WA 98195-4530, U.S.A.
  • Cesar E. SilvaDepartment of Mathematics
    Williams College
    Williamstown, MA 01267, U.S.A.
  • Jack SpielbergSchool of Mathematical
    and Statistical Sciences
    Arizona State University
    Tempe, AZ 85287-1804, U.S.A.
  • David StevensWilliams College
    Williamstown, MA 01267, U.S.A.
  • Jane WangDepartment of Mathematics
    Massachusetts Institute of Technology
    Cambridge, MA 02138, U.S.A.

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