Minimal systems and distributionally scrambled sets

Oprocha, Piotr

Bulletin de la Société Mathématique de France, Tome 140 (2012) no. 3, pp. 401-439.

In this paper we investigate numerous constructions of minimal systems from the point of view of $({ℱ}_1,{ℱ}_2)$-chaos (but most of our results concern the particular cases of distributional chaos of type $1$ and $2$). We consider standard classes of systems, such as Toeplitz flows, Grillenberger $K$-systems or Blanchard-Kwiatkowski extensions of the Chacón flow, proving that all of them are DC2. An example of DC1 minimal system with positive topological entropy is also introduced. The above mentioned results answer a few open problems known from the literature.

Classification : 37B05, 37B40, 37B10
Keywords: chaotic pair, scrambled set, Mycielski set, distributional chaos, Li-Yorke chaos, filter
Mot clés : paire chaotique, ensemble ***, ensemble de Mycielski, chaos distributionnel, chaos de Li-Yorke, filtre

[1] R. L. Adler, A. G. Konheim & M. H. Mcandrew - « Topological entropy », Trans. Amer. Math. Soc. 114 (1965), p. 309-319. | MR | Zbl

[2] E. Akin - « Lectures on Cantor and Mycielski sets for dynamical systems », in Chapel Hill Ergodic Theory Workshops, Contemp. Math., vol. 356, Amer. Math. Soc., 2004, p. 21-79. | MR | Zbl

[3] F. Balibrea, B. Schweizer, A. Sklar & J. Smítal - « Generalized specification property and distributional chaos », Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), p. 1683-1694, Dynamical systems and functional equations (Murcia, 2000). | MR | Zbl

[4] F. Balibrea & J. Smítal - « Strong distributional chaos and minimal sets », Topology Appl. 156 (2009), p. 1673-1678. | MR | Zbl

[5] F. Balibrea, J. Smítal & M. Štefánková - « The three versions of distributional chaos », Chaos Solitons Fractals 23 (2005), p. 1581-1583. | MR | Zbl

[6] J. Banks - « Regular periodic decompositions for topologically transitive maps », Ergodic Theory Dynam. Systems 17 (1997), p. 505-529. | MR | Zbl

[7] F. Blanchard, B. Host & S. Ruette - « Asymptotic pairs in positive-entropy systems », Ergodic Theory Dynam. Systems 22 (2002), p. 671-686. | MR | Zbl

[8] F. Blanchard & J. Kwiatkowski - « Minimal self-joinings and positive topological entropy. II », Studia Math. 128 (1998), p. 121-133. | MR | Zbl

[9] T. Downarowicz - « Survey of odometers and Toeplitz flows », in Algebraic and topological dynamics, Contemp. Math., vol. 385, Amer. Math. Soc., 2005, p. 7-37. | MR | Zbl

[10] H. Furstenberg - « Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation », Math. Systems Theory 1 (1967), p. 1-49. | MR | Zbl

[11] -, Recurrence in ergodic theory and combinatorial number theory, Princeton Univ. Press, 1981. | MR | Zbl

[12] C. Grillenberger - « Constructions of strictly ergodic systems. I. Given entropy », Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/73), p. 323-334. | MR | Zbl

[13] -, « Constructions of strictly ergodic systems. II. $K$-Systems », Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/73), p. 335-342. | MR | Zbl

[14] C. Grillenberger & P. Shields - « Construction of strictly ergodic systems. III. Bernoulli systems », Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33 (1975/76), p. 215-217. | MR | Zbl

[15] L. He & C. Liu - « Invariant measures and uniform positive entropy property for inverse limits », Appl. Math. J. Chinese Univ. Ser. B 14 (1999), p. 265-272, A Chinese summary appears in Gaoxiao Yingyong Shuxue Xuebao Ser. A 14 (1999), no. 3, 367. | MR | Zbl

[16] W. Huang & X. Ye - « Devaney's chaos and 2-scattering imply Li-Yorke's chaos », Topology 117 (2002), p. 259-272. | MR | Zbl

[17] W. Huang, S. Shao & X. Ye - « Mixing via sequence entropy », in Algebraic and topological dynamics, Contemp. Math., vol. 385, Amer. Math. Soc., 2005, p. 101-122. | MR | Zbl

[18] W. Huang & X. Ye - « A local variational relation and applications », Israel J. Math. 151 (2006), p. 237-279. | MR | Zbl

[19] D. Kerr & H. Li - « Independence in topological and $C^{*}$-dynamics », Math. Ann. 338 (2007), p. 869-926. | MR | Zbl

[20] P. Kurka - Topological and symbolic dynamics, Cours Spécialisés, vol. 11, Soc. Math. France, 2003. | MR | Zbl

[21] T. Y. Li & J. A. Yorke - « Period three implies chaos », Amer. Math. Monthly 82 (1975), p. 985-992. | MR | Zbl

[22] G. Liao & L. Wang - « Almost periodicity and distributional chaos », in Foundations of computational mathematics (Hong Kong, 2000), World Sci. Publ., River Edge, NJ, 2002, p. 189-210. | MR | Zbl

[23] P. Oprocha - « Distributional chaos revisited », Trans. Amer. Math. Soc. 361 (2009), p. 4901-4925. | MR | Zbl

[24] -, « Weak mixing and product recurrence », Ann. Inst. Fourier 60 (2010), p. 1233-1257. | Numdam | MR | Zbl

[25] P. Oprocha & M. Štefánková - « Specification property and distributional chaos almost everywhere », Proc. Amer. Math. Soc. 136 (2008), p. 3931-3940. | MR | Zbl

[26] P. Oprocha & P. Wilczyński - « Distributional chaos via semiconjugacy », Nonlinearity 20 (2007), p. 2661-2679. | MR | Zbl

[27] R. Pikuła - « On some notions of chaos in dimension zero », Colloq. Math. 107 (2007), p. 167-177. | MR | Zbl

[28] B. Schweizer, A. Sklar & J. Smítal - « Distributional (and other) chaos and its measurement », Real Anal. Exchange 26 (2000/01), p. 495-524. | MR | Zbl

[29] B. Schweizer & J. Smítal - « Measures of chaos and a spectral decomposition of dynamical systems on the interval », Trans. Amer. Math. Soc. 344 (1994), p. 737-754. | MR | Zbl

[30] A. Sklar & J. Smítal - « Distributional chaos on compact metric spaces via specification properties », J. Math. Anal. Appl. 241 (2000), p. 181-188. | MR | Zbl

[31] F. Tan & J. Xiong - « Chaos via Furstenberg family couple », Topology Appl. 156 (2009), p. 525-532. | MR | Zbl

[32] S. Williams - « Toeplitz minimal flows which are not uniquely ergodic », Z. Wahrsch. Verw. Gebiete 67 (1984), p. 95-107. | MR | Zbl