On $\mu $-compatible metrics and measurable sensitivity

Volume 126 / 2012

Ilya Grigoriev, Marius Cătălin Iordan, Amos Lubin, Nathaniel Ince, Cesar E. Silva Colloquium Mathematicum 126 (2012), 53-72 MSC: Primary 37A05, 37A40; Secondary 37F10. DOI: 10.4064/cm126-1-3

Abstract

We introduce the notion of W-measurable sensitivity, which extends and strictly implies canonical measurable sensitivity, a measure-theoretic version of sensitive dependence on initial conditions. This notion also implies pairwise sensitivity with respect to a large class of metrics. We show that nonsingular ergodic and conservative dynamical systems on standard spaces must be either W-measurably sensitive, or isomorphic mod 0 to a minimal uniformly rigid isometry. In the finite measure-preserving case they are W-measurably sensitive or measurably isomorphic to an ergodic isometry on a compact metric space.

Authors

  • Ilya GrigorievDepartment of Mathematics
    Stanford University
    Stanford, CA 94305, U.S.A.
    e-mail
  • Marius Cătălin IordanWilliams College
    Williamstown, MA 01267, U.S.A.
    e-mail
  • Amos LubinHarvard College
    University Hall
    Cambridge, MA 02138, U.S.A.
    e-mail
  • Nathaniel InceMassachusetts Institute of Technology
    77 Massachusetts Ave.
    Cambridge, MA 02139-4307, U.S.A.
    e-mail
  • Cesar E. SilvaDepartment of Mathematics
    Williams College
    Williamstown, MA 01267, U.S.A.
    e-mail

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