# Article

MSC: 37A25, 37A40
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Keywords:
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.
References:
[1] Aaronson, J.: An Introduction to Infinite Ergodic Theory. Mathematical Surveys and Monographs 50. American Mathematical Society, Providence, RI (1997). MR 1450400 | Zbl 0882.28013
[2] Bogachev, V. I.: Measure Theory. Vol. I. and II. Springer, Berlin (2007). MR 2267655 | Zbl 1120.28001
[3] Cadre, B., Jacob, P.: On pairwise sensitivity. J. Math. Anal. Appl. 309 (2005), 375-382. DOI 10.1016/j.jmaa.2005.01.061 | MR 2154050 | Zbl 1089.28011
[4] Helmberg, G., Simons, F. H.: Aperiodic transformations. Z. Wahrscheinlichkeitstheor. Verw. Geb. 13 (1969), 180-190. DOI 10.1007/BF00537023 | MR 0252602 | Zbl 0176.33902
[5] Huang, W., Lu, P., Ye, X.: Measure-theoretical sensitivity and equicontinuity. Isr. J. Math. 183 (2011), 233-283. DOI 10.1007/s11856-011-0049-x | MR 2811160 | Zbl 1257.37018
[6] James, J., Koberda, T., Lindsey, K., Silva, C. E., Speh, P.: Measurable sensitivity. Proc. Am. Math. Soc. 136 (2008), 3549-3559. DOI 10.1090/S0002-9939-08-09294-0 | MR 2415039 | Zbl 1149.37002
[7] Jones, L. K., Krengel, U.: On transformations without finite invariant measure. Adv. Math. 12 (1974), 275-295. DOI 10.1016/S0001-8708(74)80005-8 | MR 0340548 | Zbl 0286.28017
[8] Grigoriev, I., Iordan, M., Ince, N., Lubin, A., Silva, C. E.: On $\mu$-compatible metrics and measurable sensitivity. Colloq. Math. 126 (2012), 53-72. DOI 10.4064/cm126-1-3 | MR 2901201 | Zbl 1251.37011
[9] Kopf, C.: Negative nonsingular transformations. Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B 18 (1982), 81-102. MR 0646842 | Zbl 0482.28028
[10] Mañé, R.: Ergodic Theory and Differentiable Dynamics. Translated from Portuguese by Silvio Levy. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (1987). MR 0889254 | Zbl 0616.28007
[11] Morales, C. A.: Some properties of pairwise sensitive homeomorphisms. Preprint 2011.
[12] Parry, W.: Aperiodic transformations and generators. J. Lond. Math. Soc. 43 (1968), 191-194. DOI 10.1112/jlms/s1-43.1.191 | MR 0224774 | Zbl 0167.32901
[13] Parry, W.: Principal partitions and generators. Bull. Am. Math. Soc. 73 (1967), 307-309. DOI 10.1090/S0002-9904-1967-11727-0 | MR 0217260 | Zbl 0167.32804
[14] Parry, W.: Generators and strong generators in ergodic theory. Bull. Am. Math. Soc. 72 (1966), 294-296. DOI 10.1090/S0002-9904-1966-11498-2 | MR 0193208 | Zbl 0144.29802
[15] Steele, J. M.: Covering finite sets by ergodic images. Can. Math. Bull. 21 (1978), 85-91. DOI 10.4153/CMB-1978-013-3 | MR 0480947 | Zbl 0375.28007
[16] Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics 79. Springer, New York (1982). DOI 10.1007/978-1-4612-5775-2 | MR 0648108 | Zbl 0475.28009

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