Partition's sensitivity for measurable maps
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., 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., On pairwise sensitivity, J. Math. Anal. Appl. 309 (2005), no. 1, 375--382). Some applications are given.