In the present paper, we consider the problem of existence of nonequivalent definitions of topological transitivity, which is a classical problem of the topological dynamics. In particular, we use the fact that all available definitions of this kind imply a condition imposed on the dynamical system. The main result of our investigations is the complete classification of these dynamical systems.
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References
E. Akin, The General Topology of Dynamical Systems, Graduate Studies in Mathematics, 1, American Mathematical Society, Providence (1993).
E. Akin, Recurrence in Topological Dynamics: Furstenberg Families and Ellis Actions, Plenum, New York (1997).
E. Akin and J. D. Carlson, “Conceptions of topological transitivity,” Topology Appl., 159, 2815–2830 (2012).
J. Banks, “Topological mapping properties defined by digraphs,” Discrete Contin. Dynam. Systems, 5, 83–92 (1999).
J. Banks and B. Stanley, “A note on equivalent definitions of topological transitivity,” Discrete Contin. Dynam. Systems, Ser. A, 33, 293–1296 (2013).
S. Kolyada and L. Snoha, “Some aspects of topological transitivity — a survey,” Grazer Math. Berichte., 334, 3–35 (1997).
J.-H. Mai and W.-H. Sun, “Transitivities of maps of general topological spaces,” Topology Appl., 157, 353–375 (2010).
S. Silverman, “On maps with dense orbits and the definition of chaos,” Rocky Mountain J. Math., 22, 353–375 (1992).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 9, pp. 1163–1185, September, 2013.
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Bilokopytov, E., Kolyada, S.F. Transitive Maps on Topological Spaces. Ukr Math J 65, 1293–1318 (2014). https://doi.org/10.1007/s11253-014-0860-8
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DOI: https://doi.org/10.1007/s11253-014-0860-8