On \mu-Compatible Metrics and Measurable Sensitivity

TL;DR

This paper introduces W-measurable sensitivity, a new measure-theoretic concept that characterizes the behavior of dynamical systems, distinguishing between sensitive and rigid systems in a broad class of measure spaces.

Contribution

The paper defines W-measurable sensitivity, extends the concept of measurable sensitivity, and characterizes ergodic systems as either sensitive or isomorphic to rigid isometries.

Findings

Nonsingular ergodic systems are either W-measurably sensitive or isomorphic to minimal rigid isometries.

Finite measure-preserving systems are either W-measurably sensitive or measure-theoretically isomorphic to ergodic isometries.

W-measurable sensitivity strictly implies canonical measurable sensitivity, providing a hierarchy of sensitivity notions.

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.