C. A. Morales, Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970, Rio de Janeiro, Brazil, e-mail: morales@impa.br
Abstract: We study countable partitions for measurable maps on measure spaces such that, for every point $x$, the set of points with the same itinerary as that of $x$ is negligible. We prove in nonatomic probability spaces that every strong generator (Parry, W., 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 ergodic measure-preserving ones with positive entropy on probability spaces (thus extending the result in Cadre, B., Jacob, P., On pairwise sensitivity, J. Math. Anal. Appl. 309 (2005), 375-382). Some applications are given.
Keywords: measurable map, measure space, expansive map
Classification (MSC 2010): 37A25, 37A40
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