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Expansivity for measures on uniform spaces


Authors: C. A. Morales and V. Sirvent
Journal: Trans. Amer. Math. Soc. 368 (2016), 5399-5414
MSC (2010): Primary 54H20; Secondary 54E15
DOI: https://doi.org/10.1090/tran/6555
Published electronically: November 6, 2015
MathSciNet review: 3458385
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Abstract: We define positively expansive and expansive measures on uniform spaces extending the analogous concepts on metric spaces. We show that such measures can exist for measurable or bimeasurable maps on compact non-Hausdorff uniform spaces. We prove that positively expansive probability measures on Lindelöf spaces are non-atomic and their corresponding maps eventually aperiodic. We prove that the stable classes of measurable maps have measure zero with respect to any positively expansive invariant measure. In addition, any measurable set where a measurable map in a Lindelöf uniform space is Lyapunov stable has measure zero with respect to any positively expansive inner regular measure. We conclude that the set of sinks of any bimeasurable map with canonical coordinates of a Lindelöf space has zero measure with respect to any positively expansive inner regular measure. Finally, we show that every measurable subset of points with converging semiorbits of a bimeasurable map on a separable uniform space has zero measure with respect to every expansive outer regular measure. These results generalize those found in works by Arbieto and Morales and by Reddy and Robertson.


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  • [1] A. Arbieto and C. A. Morales, Some properties of positive entropy maps, Ergodic Theory Dynam. Systems 34 (2014), no. 3, 765-776. MR 3199792, https://doi.org/10.1017/etds.2012.162
  • [2] M. Awartani and S. Elaydi, An extension of chaotic dynamics to general topological spaces, Panamer. Math. J. 10 (2000), no. 2, 61-71. MR 1754512 (2000m:37016)
  • [3] V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655 (2008g:28002)
  • [4] M. Brin and A. Katok, On local entropy, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 30-38. MR 730261 (85c:58063), https://doi.org/10.1007/BFb0061408
  • [5] Thomas A. Brown and W. W. Comfort, New method for expansion and contraction maps in uniform spaces, Proc. Amer. Math. Soc. 11 (1960), 483-486. MR 0113210 (22 #4048)
  • [6] B. F. Bryant, On expansive homeomorphisms, Pacific J. Math. 10 (1960), 1163-1167. MR 0120632 (22 #11382)
  • [7] Benoît Cadre and Pierre Jacob, On pairwise sensitivity, J. Math. Anal. Appl. 309 (2005), no. 1, 375-382. MR 2154050 (2006b:28026), https://doi.org/10.1016/j.jmaa.2005.01.061
  • [8] Tullio Ceccherini-Silberstein and Michel Coornaert, Sensitivity and Devaney's chaos in uniform spaces, J. Dyn. Control Syst. 19 (2013), no. 3, 349-357. MR 3085695, https://doi.org/10.1007/s10883-013-9182-7
  • [9] Tarun Das, Keonhee Lee, David Richeson, and Jim Wiseman, Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces, Topology Appl. 160 (2013), no. 1, 149-158. MR 2995088, https://doi.org/10.1016/j.topol.2012.10.010
  • [10] Ruchi Das and J. R. Patadia, Expansive homeomorphisms on topological spaces, J. Indian Math. Soc. (N.S.) 65 (1998), no. 1-4, 211-218. MR 1750373 (2001d:54033)
  • [11] Murray Eisenberg, Expansive transformation semigroups of endomorphisms, Fund. Math. 59 (1966), 313-321. MR 0203713 (34 #3562)
  • [12] Marcus B. Feldman, A proof of Lusin's theorem, Amer. Math. Monthly 88 (1981), no. 3, 191-192. MR 619565 (82h:28006), https://doi.org/10.2307/2320466
  • [13] Walter Helbig Gottschalk and Gustav Arnold Hedlund, Topological dynamics, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R. I., 1955. MR 0074810 (17,650e)
  • [14] B. M. Hood, Topological entropy and uniform spaces, J. London Math. Soc. (2) 8 (1974), 633-641. MR 0353282 (50 #5766)
  • [15] I. M. James, Topological and uniform spaces, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1987. MR 884154 (89b:54001)
  • [16] Takashi Kimura, Completion theorem for uniform entropy, Comment. Math. Univ. Carolin. 39 (1998), no. 2, 389-399. MR 1651987 (2000g:54035)
  • [17] J. D. Knowles, On the existence of non-atomic measures, Mathematika 14 (1967), 62-67. MR 0214719 (35 #5568)
  • [18] Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8, Springer-Verlag, Berlin, 1987. Translated from the Portuguese by Silvio Levy. MR 889254 (88c:58040)
  • [19] C. A. Morales, Partition sensitivity for measurable maps, Math. Bohem. 138 (2013), no. 2, 133-148. MR 3112360
  • [20] Carlos Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst. 32 (2012), no. 1, 293-301. MR 2837062 (2012k:37027), https://doi.org/10.3934/dcds.2012.32.293
  • [21] W. Parry, Aperiodic transformations and generators, J. London Math. Soc. 43 (1968), 191-194. MR 0224774 (37 #373)
  • [22] William Reddy, The existence of expansive homeomorphisms on manifolds, Duke Math. J. 32 (1965), 627-632. MR 0187226 (32 #4679)
  • [23] William L. Reddy and Lewis C. Robertson, Sources, sinks and saddles for expansive homeomorphisms with canonical coordinates, Rocky Mountain J. Math. 17 (1987), no. 4, 673-681. MR 923738 (89g:54095), https://doi.org/10.1216/RMJ-1987-17-4-673
  • [24] Walter Rudin, Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc. 8 (1957), 39-42. MR 0085475 (19,46b)
  • [25] W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769-774. MR 0038022 (12,344b)
  • [26] A. Weil, Sur les espaces á structure uniforme et sur la topologie générale, Act. Sci. Ind. 551, Paris, 1937.

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Additional Information

C. A. Morales
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, 21945-970 Rio de Janeiro, Brazil
Email: morales@impa.br

V. Sirvent
Affiliation: Departamento de Matemáticas, Universidad Simón Bolívar, Apartado 89000, Caracas 1086-A, Venezuela
Email: vsirvent@usb.ve

DOI: https://doi.org/10.1090/tran/6555
Keywords: Expansive measure, uniform space, Borel measure
Received by editor(s): January 9, 2013
Received by editor(s) in revised form: May 21, 2014, and June 25, 2014
Published electronically: November 6, 2015
Additional Notes: The first author was partially supported by CNPq, FAPERJ and PRONEX/DS from Brazil.
Article copyright: © Copyright 2015 American Mathematical Society

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