On μ-Compatible Metrics and Measurable Sensitivity

Details On μ-Compatible Metrics and Measurable Sensitivity On μ-Compatible Metrics and Measurable Sensitivity

2009-10-10

arxiv.org/abs/0910.1958v2

Mathematics

Dynamical Systems

Many improvements in exposition, a technical assumption removed, as suggested by the reviewer

Scientific paper

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.

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