On $\mu $-compatible metrics and measurable sensitivity
Volume 126 / 2012
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.
