Partition's sensitivity for measurable maps

TL;DR

This paper investigates the properties of countable partitions in measurable maps, characterizing when they are strong generators and their relation to aperiodicity, with implications for ergodic theory and entropy.

Contribution

It provides a characterization of partitions with negligible itinerary sets as strong generators in nonatomic probability spaces, extending previous results on ergodic maps with positive entropy.

Findings

Strong generators satisfy the negligible itinerary property in nonatomic spaces.

Maps with these partitions are aperiodic and nonatomic.

Includes ergodic measure-preserving maps with positive entropy.

Abstract

We study countable partitions for measurable maps on measure spaces such that for all point $x$ the set of points with the same itinerary of $x$ is negligible. We prove that in nonatomic probability spaces every strong generator (Parry, W., {\em Aperiodic transformations and generators}, J. London Math. Soc. 43 (1968), 191--194) satisfies this property but not conversely. In addition, measurable maps carrying partitions with this property are aperiodic and their corresponding spaces are nonatomic. From this we obtain a characterization of nonsingular countable to one mappings with these partitions on nonatomic Lebesgue probability spaces as those having strong generators. Furthermore, maps carrying these partitions include the ergodic measure-preserving ones with positive entropy on probability spaces (thus extending a result in Cadre, B., Jacob, P., {\em On pairwise sensitivity}, J. Math. Anal. Appl. 309 (2005), no. 1, 375--382). Some applications are given.