Geodesic
Equicontinuity, shadowing and distality in general topological spaces
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 711-726
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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.
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.
DOI : 10.21136/CMJ.2020.0488-18
Classification : 37B05, 37B20, 54H20
Keywords: shadowing; chain transitive; equicontinuity; uniform space 
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Wang, Huoyun. Equicontinuity, shadowing and distality in general topological spaces. Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 711-726. doi: 10.21136/CMJ.2020.0488-18

[1] Akin, E., Auslander, J., Berg, K.: 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 | MR | JFM

[2] Akin, E., Kolyada, S.: Li-Yorke sensitivity. Nonlinearity 16 (2003), 1421-1433. | DOI | MR | JFM

[3] Auslander, J., Greschonig, G., Nagar, A.: Reflections on equicontinuity. Proc. Am. Math. Soc. 142 (2014), 3129-3137. | DOI | MR | JFM

[4] Auslander, J., Yorke, J. A.: Interval maps, factors of maps, and chaos. Tohoku Math. J., II. Ser. 32 (1980), 177-188. | DOI | MR | JFM

[5] Bergelson, V.: 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 | MR | JFM

[6] Blanchard, F., Glasner, E., Kolyada, S., Maass, A.: On Li-York pairs. J. Reine Angew. Math. 547 (2002), 51-68. | DOI | MR | JFM

[7] Brian, W.: Abstract $\omega$-limit sets. J. Symb. Log. 83 (2018), 477-495. | DOI | MR | JFM

[8] Ceccherini-Silberstein, T., Coornaert, M.: Sensitivity and Devaney's chaos in uniform spaces. J. Dyn. Control Syst. 19 (2013), 349-357. | DOI | MR | JFM

[9] Das, P., Das, T.: Various types of shadowing and specification on uniform spaces. J. Dyn. Control Syst. 24 (2018), 253-267. | DOI | MR | JFM

[10] Dastjerdi, D. A., Hosseini, M.: Shadowing with chain transitivity. Topology Appl. 156 (2009), 2193-2195. | DOI | MR | JFM

[11] Ellis, D., Ellis, R., Nerurkar, M.: The topological dynamics of semigroup actions. Trans. Am. Math. Soc. 353 (2001), 1279-1320. | DOI | MR | JFM

[12] Engelking, R.: General Topology. Sigma Series in Pure Mathematics 6, Heldermann, Berlin (1989). | MR | JFM

[13] Glasner, E., Weiss, B.: Sensitive dependence on initial conditions. Nonlinearity 6 (1993), 1067-1075. | DOI | MR | JFM

[14] Good, C., Macías, S.: What is topological about topological dynamics?. Discrete Contin. Dyn. Syst. 38 (2018), 1007-1031. | DOI | MR | JFM

[15] Hindman, N., Strauss, D.: Algebra in the Stone-Čech Compactification: Theory and Applications. De Gruyter Expositions in Mathematics 27, Walter de Gruyter, Berlin (1998). | DOI | MR | JFM

[16] Hood, B. M.: Topological entropy and uniform spaces. J. Lond. Math. Soc., II. Ser. 8 (1974), 633-641. | DOI | MR | JFM

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