| Title:
|
Partition sensitivity for measurable maps (English) |
| Author:
|
Morales, C. A. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
138 |
| Issue:
|
2 |
| Year:
|
2013 |
| Pages:
|
133-148 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
measurable map |
| Keyword:
|
measure space |
| Keyword:
|
expansive map |
| MSC:
|
37A25 |
| MSC:
|
37A40 |
| idZBL:
|
Zbl 06221244 |
| idMR:
|
MR3112360 |
| DOI:
|
10.21136/MB.2013.143286 |
| . |
| Date available:
|
2013-05-27T14:21:50Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143286 |
| . |
| Reference:
|
[1] Aaronson, J.: An Introduction to Infinite Ergodic Theory.Mathematical Surveys and Monographs 50. American Mathematical Society, Providence, RI (1997). Zbl 0882.28013, MR 1450400, 10.1090/surv/050 |
| Reference:
|
[2] Bogachev, V. I.: Measure Theory. Vol. I. and II.Springer, Berlin (2007). Zbl 1120.28001, MR 2267655 |
| Reference:
|
[3] Cadre, B., Jacob, P.: On pairwise sensitivity.J. Math. Anal. Appl. 309 (2005), 375-382. Zbl 1089.28011, MR 2154050, 10.1016/j.jmaa.2005.01.061 |
| Reference:
|
[4] Helmberg, G., Simons, F. H.: Aperiodic transformations.Z. Wahrscheinlichkeitstheor. Verw. Geb. 13 (1969), 180-190. Zbl 0176.33902, MR 0252602, 10.1007/BF00537023 |
| Reference:
|
[5] Huang, W., Lu, P., Ye, X.: Measure-theoretical sensitivity and equicontinuity.Isr. J. Math. 183 (2011), 233-283. Zbl 1257.37018, MR 2811160, 10.1007/s11856-011-0049-x |
| Reference:
|
[6] James, J., Koberda, T., Lindsey, K., Silva, C. E., Speh, P.: Measurable sensitivity.Proc. Am. Math. Soc. 136 (2008), 3549-3559. Zbl 1149.37002, MR 2415039, 10.1090/S0002-9939-08-09294-0 |
| Reference:
|
[7] Jones, L. K., Krengel, U.: On transformations without finite invariant measure.Adv. Math. 12 (1974), 275-295. Zbl 0286.28017, MR 0340548, 10.1016/S0001-8708(74)80005-8 |
| Reference:
|
[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. Zbl 1251.37011, MR 2901201, 10.4064/cm126-1-3 |
| Reference:
|
[9] Kopf, C.: Negative nonsingular transformations.Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B 18 (1982), 81-102. Zbl 0482.28028, MR 0646842 |
| Reference:
|
[10] Mañé, R.: Ergodic Theory and Differentiable Dynamics.Translated from Portuguese by Silvio Levy. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (1987). Zbl 0616.28007, MR 0889254 |
| Reference:
|
[11] Morales, C. A.: Some properties of pairwise sensitive homeomorphisms.Preprint 2011. |
| Reference:
|
[12] Parry, W.: Aperiodic transformations and generators.J. Lond. Math. Soc. 43 (1968), 191-194. Zbl 0167.32901, MR 0224774, 10.1112/jlms/s1-43.1.191 |
| Reference:
|
[13] Parry, W.: Principal partitions and generators.Bull. Am. Math. Soc. 73 (1967), 307-309. Zbl 0167.32804, MR 0217260, 10.1090/S0002-9904-1967-11727-0 |
| Reference:
|
[14] Parry, W.: Generators and strong generators in ergodic theory.Bull. Am. Math. Soc. 72 (1966), 294-296. Zbl 0144.29802, MR 0193208, 10.1090/S0002-9904-1966-11498-2 |
| Reference:
|
[15] Steele, J. M.: Covering finite sets by ergodic images.Can. Math. Bull. 21 (1978), 85-91. Zbl 0375.28007, MR 0480947, 10.4153/CMB-1978-013-3 |
| Reference:
|
[16] Walters, P.: An Introduction to Ergodic Theory.Graduate Texts in Mathematics 79. Springer, New York (1982). Zbl 0475.28009, MR 0648108, 10.1007/978-1-4612-5775-2 |
| . |
