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A characterization of $ \mu-$equicontinuity for topological dynamical systems

Author: Felipe García-Ramos
Journal: Proc. Amer. Math. Soc. 145 (2017), 3357-3368
MSC (2010): Primary 37B05, 37A50, 54H20, 28A75
Published electronically: April 26, 2017
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Abstract: Two different notions of measure theoretical equicontinuity
($ \mu -$equicontinuity) for topological dynamical systems with respect to Borel probability measures appeared in works by Gilman (1987) and Huang, Lee and Ye (2011). We show that if the probability space satisfies Lebesgue's density theorem and Vitali's covering theorem (for example a Cantor set or a subset of $ \mathbb{R}^{d}$), then both notions are equivalent. To show this we characterize Lusin measurable maps using $ \mu -$continuity points. As a corollary we also obtain a new characterization of $ \mu -$mean equicontinuity.

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  • [1] Ethan Akin, Joseph Auslander, and Kenneth Berg, When is a transitive map chaotic?, Convergence in ergodic theory and probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., vol. 5, de Gruyter, Berlin, 1996, pp. 25-40. MR 1412595
  • [2] Joseph Auslander and James A. Yorke, Interval maps, factors of maps, and chaos, Tôhoku Math. J. (2) 32 (1980), no. 2, 177-188. MR 580273,
  • [3] Louis Block and James Keesling, A characterization of adding machine maps, Topology Appl. 140 (2004), no. 2-3, 151-161. MR 2074913,
  • [4] A. M. Bruckner and Thakyin Hu, Equicontinuity of iterates of an interval map, Tamkang J. Math. 21 (1990), no. 3, 287-294. MR 1079314
  • [5] Benoît Cadre and Pierre Jacob, On pairwise sensitivity, J. Math. Anal. Appl. 309 (2005), no. 1, 375-382. MR 2154050,
  • [6] Tomasz Downarowicz, Survey of odometers and Toeplitz flows, Algebraic and topological dynamics, Contemp. Math., vol. 385, Amer. Math. Soc., Providence, RI, 2005, pp. 7-37. MR 2180227,
  • [7] T. Downarowicz and E. Glasner, Isomorphic extensions and applications, arXiv preprint arXiv:1502.06999 5 (2015).
  • [8] Rick Durrett, Probability: theory and examples, 4th ed., Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010. MR 2722836
  • [9] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • [10] S. Fomin, On dynamical systems with a purely point spectrum, Doklady Akad. Nauk SSSR (N.S.) 77 (1951), 29-32 (Russian). MR 0043397
  • [11] Felipe García-Ramos, Limit behaviour of $ \mu $-equicontinuous cellular automata, Theoret. Comput. Sci. 623 (2016), 2-14. MR 3476561,
  • [12] F. García-Ramos, Weak forms of topological and measure theoretical equicontinuity: relationships with discrete spectrum and sequence entropy, Ergodic Theory and Dynamical Systems (available on CJO2016. doi:10.1017/etds.2015.83 (2016).
  • [13] Robert H. Gilman, Periodic behavior of linear automata, Dynamical systems (College Park, MD, 1986-87) Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 216-219. MR 970557,
  • [14] Robert H. Gilman, Classes of linear automata, Ergodic Theory Dynam. Systems 7 (1987), no. 1, 105-118. MR 886373,
  • [15] Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917
  • [16] Wen Huang, Ping Lu, and Xiangdong Ye, Measure-theoretical sensitivity and equicontinuity, Israel J. Math. 183 (2011), 233-283. MR 2811160,
  • [17] A. Käenmäki, T. Rajala, and V. Suomala, Local homogeneity and dimensions of measures, arXiv preprint arXiv:1003.2895 (2010).
  • [18] R. Kannan and Carole King Krueger, Advanced analysis on the real line, Universitext, Springer-Verlag, New York, 1996. MR 1390758
  • [19] Jian Li, Siming Tu, and Xiangdong Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory Dynam. Systems 35 (2015), no. 8, 2587-2612. MR 3456608
  • [20] Douglas Lind and Brian Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. MR 1369092
  • [21] Michel Simonnet, Measures and probabilities, Universitext, Springer-Verlag, New York, 1996. With a foreword by Charles-Michel Marle. MR 1416330
  • [22] Antonios Valaristos, Equicontinuity of iterates of circle maps, Internat. J. Math. Math. Sci. 21 (1998), no. 3, 453-458. MR 1620310,
  • [23] Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108

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Additional Information

Felipe García-Ramos
Affiliation: Catedras Conacyt, Av. Insurgentes Sur 1582, Benito Juarez CDMX, 03940, Mexico – and – Instituto de Física, Universidad Autónoma de San Luis Potosí, Av. Manuel Nava #6, Zona Universitaria San Luis Potosí, SLP, 78290, México

Received by editor(s): July 31, 2014
Received by editor(s) in revised form: August 17, 2015, May 13, 2016, and July 28, 2016
Published electronically: April 26, 2017
Additional Notes: While writing and correcting this paper the author was supported by IMPA, CAPES and NSERC. The author would like to thank Brian Marcus for his support and comments.
Communicated by: Nimish Shah
Article copyright: © Copyright 2017 American Mathematical Society

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