On Li-Yorke Measurable Sensitivity

TL;DR

This paper introduces a measurable version of Li-Yorke sensitivity for nonsingular dynamical systems, establishing its relationship with mixing properties and showing its equivalence to weak mixing in finite measure cases.

Contribution

It defines Li-Yorke measurable sensitivity for nonsingular systems and explores its implications and equivalences with mixing properties, extending the understanding of sensitivity in ergodic theory.

Findings

Ergodic Cartesian square implies Li-Yorke measurable sensitivity.

Li-Yorke measurable sensitivity implies weak mixing.

In finite measure-preserving systems, Li-Yorke measurable sensitivity is equivalent to weak mixing.

Abstract

The notion of Li-Yorke sensitivity has been studied extensively in the case of topological dynamical systems. We introduce a measurable version of Li-Yorke sensitivity, for nonsingular (and measure-preserving) dynamical systems, and compare it with various mixing notions. It is known that in the case of nonsingular dynamical systems, ergodic Cartesian square implies double ergodicity, which in turn implies weak mixing, but the converses do not hold in general, though they are all equivalent in the finite measure-preserving case. We show that for nonsingular systems, ergodic Cartesian square implies Li-Yorke measurable sensitivity, which in turn implies weak mixing. As a consequence we obtain that, in the finite measure-preserving case, Li-Yorke measurable sensitivity is equivalent to weak mixing. We also show that with respect to totally bounded metrics, double ergodicity implies Li-Yorke measurable sensitivity.