Foryś-Krawiec, Magdalena; Oprocha, Piotr; Štefánková, Marta Distributionally chaotic systems of type 2 and rigidity. (English) Zbl 1376.37022 J. Math. Anal. Appl. 452, No. 1, 659-672 (2017). Summary: In this paper we deal with uniformly rigid systems obtained by a method introduced by Y. Katznelson and B. Weiss [Prog. Math. 10, 195–210 (1981; Zbl 0469.54023)] and show that such systems never contain DC2 pairs. On the other hand, we introduce a modification of this technique that leads to a uniformly rigid system with DC2 pairs. We also show that every dynamical system contains a pair of distinct points which is not DC2. Cited in 1 Document MSC: 37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) 37A25 Ergodicity, mixing, rates of mixing 54H20 Topological dynamics (MSC2010) Keywords:uniformly rigid; distributional chaos; Hilbert cube Citations:Zbl 0469.54023 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Akin, Ethan; Auslander, Joseph; Berg, Kenneth, When is a transitive map chaotic?, (Convergence in Ergodic Theory and Probability. Convergence in Ergodic Theory and Probability, Columbus, OH, 1993. Convergence in Ergodic Theory and Probability. Convergence in Ergodic Theory and Probability, Columbus, OH, 1993, Ohio State Univ. Math. Res. Inst. Publ., vol. 5 (1996), de Gruyter: de Gruyter Berlin), 25-40 · Zbl 0861.54034 [2] Blanchard, F.; Host, B.; Ruette, S., Asymptotic pairs in positive-entropy systems, Ergodic Theory Dynam. Systems, 22, 3, 671-686 (2002) · Zbl 1018.37005 [3] Downarowicz, T., Positive topological entropy implies chaos DC2, Proc. Amer. Math. Soc., 142, 1, 137-149 (2014) · Zbl 1304.37008 [4] Foryś, Magdalena; Huang, Wen; Li, Jian; Oprocha, Piotr, Invariant scrambled sets, uniform rigidity and weak mixing, Israel J. Math., 211, 1, 447-472 (2016) · Zbl 1359.37021 [5] Furstenberg, H., Recurrence in Ergodic Theory and Combinatorial Number Theory (1981), Princeton University Press: Princeton University Press Princeton, N.J., M. B. Porter Lectures · Zbl 0459.28023 [6] Furstenberg, Harry, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Syst. Theory, 1, 1-49 (1967) · Zbl 0146.28502 [7] Garcia-Ramos, Felipe; Jin, Lei, Mean proximality and mean Li-Yorke chaos · Zbl 1375.37107 [8] Glasner, S.; Maon, D., Rigidity in topological dynamics, Ergodic Theory Dynam. Systems, 9, 2, 309-320 (1989) · Zbl 0661.58027 [9] Huang, Wen; Ye, Xiangdong, Homeomorphisms with the whole compacta being scrambled sets, Ergodic Theory Dynam. Systems, 21, 1, 77-91 (2001) · Zbl 0978.37003 [10] Huang, Wen; Ye, Xiangdong, Minimal sets in almost equicontinuous systems, Tr. Mat. Inst. Steklova, 244, 297-304 (2004) · Zbl 1072.37012 [11] Katznelson, Yitzhak; Weiss, Benjamin, When all points are recurrent/generic, (Ergodic Theory and Dynamical Systems, I. Ergodic Theory and Dynamical Systems, I, College Park, Md., 1979-80. Ergodic Theory and Dynamical Systems, I. Ergodic Theory and Dynamical Systems, I, College Park, Md., 1979-80, Progr. Math., vol. 10 (1981), Birkhäuser: Birkhäuser Boston, Mass.), 195-210 · Zbl 0469.54023 [12] Kulczycki, Marcin; Kwietniak, Dominik; Oprocha, Piotr, On almost specification and average shadowing properties, Fund. Math., 224, 3, 241-278 (2014) · Zbl 1352.37068 [13] Li, Jian; Tu, Siming, On proximality with Banach density one, J. Math. Anal. Appl., 416, 1, 36-51 (2014) · Zbl 1317.37026 [14] Oprocha, Piotr, Distributional chaos revisited, Trans. Amer. Math. Soc., 361, 9, 4901-4925 (2009) · Zbl 1179.37017 [15] Oprocha, Piotr; Wilczyński, Paweł, A study of chaos for processes under small perturbations, Publ. Math. Debrecen, 76, 1-2, 101-116 (2010) · Zbl 1224.34135 [16] Pikuła, Rafał, On some notions of chaos in dimension zero, Colloq. Math., 107, 2, 167-177 (2007) · Zbl 1130.37327 [17] Schweizer, B.; Smítal, J., Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344, 2, 737-754 (1994) · Zbl 0812.58062 [18] Srivastava, S. M., A Course on Borel Sets, Graduate Texts in Mathematics, vol. 180 (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0903.28001 [19] Walters, Peter, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79 (1982), Springer-Verlag: Springer-Verlag New York-Berlin · Zbl 0475.28009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.