Partition's sensitivity for measurable maps

C. A. Morales

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arXiv · 1111.2820

Partition's sensitivity for measurable maps

C. A. Morales

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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.

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C. A. Morales. 2011-11-11. Partition's sensitivity for measurable maps. https://arxiv.org/abs/1111.2820

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