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Keywords:
measurable map; measure space; expansive map
measurable map; measure space; expansive map
Summary:
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.
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