Czechoslovak Mathematical Journal, Vol. 70, No. 3, pp. 711-726, 2020


Equicontinuity, shadowing and distality in general topological spaces

Huoyun Wang

Received November 4, 2018.   Published online January 16, 2020.

Abstract:  We consider the notions of equicontinuity point, sensitivity point and so on from a topological point of view. Many of these notions can be sensibly defined either in terms of (finite) open covers or uniformities. We show that for the notions of equicontinuity point and sensitivity point, Hausdorff or uniform versions coincide in compact Hausdorff spaces and are equivalent to the standard definitions stated in terms of a metric in compact metric spaces. We prove that a uniformly chain transitive map with uniform shadowing property on a compact Hausdorff uniform space is either uniformly equicontinuous or it has no uniform equicontinuity points.
Keywords:  shadowing; chain transitive; equicontinuity; uniform space
Classification MSC:  37B20, 37B05, 54H20
DOI:  10.21136/CMJ.2020.0488-18

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References:
[1] E. Akin, J. Auslander, K. Berg: When is a transitive map chaotic? Convergence in Ergodic Theory and Probability. Ohio State University Mathematical Research Institute Publications 5, de Gruyter, Berlin (1996), 25-40. DOI 10.1515/9783110889383 | MR 1412595 | Zbl 0861.54034
[2] E. Akin, S. Kolyada: Li-Yorke sensitivity. Nonlinearity 16 (2003), 1421-1433. DOI 10.1088/0951-7715/16/4/313 | MR 1986303 | Zbl 1045.37004
[3] J. Auslander, G. Greschonig, A. Nagar: Reflections on equicontinuity. Proc. Am. Math. Soc. 142 (2014), 3129-3137. DOI 10.1090/S0002-9939-2014-12034-X | MR 3223369 | Zbl 1327.37005
[4] J. Auslander, J. A. Yorke: Interval maps, factors of maps, and chaos. Tohoku Math. J., II. Ser. 32 (1980), 177-188. DOI 10.2748/tmj/1178229634 | MR 580273 | Zbl 0448.54040
[5] V. Bergelson: Minimal idempotents and ergodic Ramsey theory. Topics in Dynamics and Ergodic Theory. London Mathematical Society Lecture Note Series 310, Cambridge University Press, Cambridge (2003), 8-39. DOI 10.1017/CBO9780511546716.004 | MR 2052273 | Zbl 1039.05063
[6] F. Blanchard, E. Glasner, S. Kolyada, A. Maass: On Li-York pairs. J. Reine Angew. Math. 547 (2002), 51-68. DOI 10.1515/crll.2002.053 | MR 1900136 | Zbl 1059.37006
[7] W. Brian: Abstract $\omega$-limit sets. J. Symb. Log. 83 (2018), 477-495. DOI 10.1017/jsl.2018.11 | MR 3835074 | Zbl 1406.54020
[8] T. Ceccherini-Silberstein, M. Coornaert: Sensitivity and Devaney's chaos in uniform spaces. J. Dyn. Control Syst. 19 (2013), 349-357. DOI 10.1007/s10883-013-9182-7 | MR 3085695 | Zbl 1338.37015
[9] P. Das, T. Das: Various types of shadowing and specification on uniform spaces. J. Dyn. Control Syst. 24 (2018), 253-267. DOI 10.1007/s10883-017-9388-1 | MR 3769323 | Zbl 1385.37013
[10] D. A. Dastjerdi, M. Hosseini: Shadowing with chain transitivity. Topology Appl. 156 (2009), 2193-2195. DOI 10.1016/j.topol.2009.04.021 | MR 2544125 | Zbl 1178.54018
[11] D. Ellis, R. Ellis, M. Nerurkar: The topological dynamics of semigroup actions. Trans. Am. Math. Soc. 353 (2001), 1279-1320. DOI 10.1090/S0002-9947-00-02704-5 | MR 1806740 | Zbl 0976.54039
[12] R. Engelking: General Topology. Sigma Series in Pure Mathematics 6, Heldermann, Berlin (1989). MR 1039321 | Zbl 0684.54001
[13] E. Glasner, B. Weiss: Sensitive dependence on initial conditions. Nonlinearity 6 (1993), 1067-1075. DOI 10.1088/0951-7715/6/6/014 | MR 1251259 | Zbl 0790.58025
[14] C. Good, S. Macías: What is topological about topological dynamics? Discrete Contin. Dyn. Syst. 38 (2018), 1007-1031. DOI 10.3934/dcds.2018043 | MR 3808985 | Zbl 1406.54021
[15] N. Hindman, D. Strauss: Algebra in the Stone-Čech Compactification: Theory and Applications. De Gruyter Expositions in Mathematics 27, Walter de Gruyter, Berlin (1998). DOI 10.1515/9783110809220 | MR 1642231 | Zbl 0918.22001
[16] B. M. Hood: Topological entropy and uniform spaces. J. Lond. Math. Soc., II. Ser. 8 (1974), 633-641. DOI 10.1112/jlms/s2-8.4.633 | MR 0353282 | Zbl 0291.54051

Affiliations:   Huoyun Wang, Department of Mathematics of Guangzhou University, 230 Wai Huan Xi Road, Guangzhou Higher Education Mega Center, Guangzhou 510006, P. R. China, e-mail: wanghuoyun@126.com


 
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