A characterization of {\mu}-equicontinuity for topological dynamical systems

TL;DR

This paper investigates two notions of {}equicontinuity} in topological dynamical systems, establishing their equivalence under certain measure-theoretic conditions and characterizing Lusin measurability.

Contribution

It proves the equivalence of two {}equicontinuity notions under Lebesgue density and Vitali covering conditions, and characterizes Lusin measurable functions.

Findings

The two notions of {}equicontinuity are equivalent under specified measure conditions.

Characterization of when a function is Lusin measurable.

Application to systems on Cantor sets or subsets of R^d.

Abstract

Two different notions of {\mu}-equicontinuity that apply to topological dynamical systems and probability measures were studied by Gilman (1987) and Huang-Lu-Ye (2011). One was used to classify measure preserving topological dynamical systems and the other for cellular automata. We show that if the probability space satisfies Lebesgue's density theorem and Vitali's covering theorem (for example a Cantor set or a subset of R^{d}) then both properties are equivalent. To do this we characterize when a function is Lusin measurable.