Minimal systems and distributionally scrambled sets

Piotr Oprocha

Bulletin de la Société Mathématique de France (2012)

  • Volume: 140, Issue: 3, page 401-439
  • ISSN: 0037-9484

Abstract

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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.

How to cite

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Oprocha, Piotr. "Minimal systems and distributionally scrambled sets." Bulletin de la Société Mathématique de France 140.3 (2012): 401-439. <http://eudml.org/doc/272610>.

@article{Oprocha2012,
abstract = {In this paper we investigate numerous constructions of minimal systems from the point of view of $(\mathcal \{F\}_1,\mathcal \{F\}_2)$-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.},
author = {Oprocha, Piotr},
journal = {Bulletin de la Société Mathématique de France},
keywords = {chaotic pair; scrambled set; Mycielski set; distributional chaos; Li-Yorke chaos; filter},
language = {eng},
number = {3},
pages = {401-439},
publisher = {Société mathématique de France},
title = {Minimal systems and distributionally scrambled sets},
url = {http://eudml.org/doc/272610},
volume = {140},
year = {2012},
}

TY - JOUR
AU - Oprocha, Piotr
TI - Minimal systems and distributionally scrambled sets
JO - Bulletin de la Société Mathématique de France
PY - 2012
PB - Société mathématique de France
VL - 140
IS - 3
SP - 401
EP - 439
AB - In this paper we investigate numerous constructions of minimal systems from the point of view of $(\mathcal {F}_1,\mathcal {F}_2)$-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.
LA - eng
KW - chaotic pair; scrambled set; Mycielski set; distributional chaos; Li-Yorke chaos; filter
UR - http://eudml.org/doc/272610
ER -

References

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  1. [1] R. L. Adler, A. G. Konheim & M. H. McAndrew – « Topological entropy », Trans. Amer. Math. Soc.114 (1965), p. 309–319. Zbl0127.13102MR175106
  2. [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. Zbl1064.37015MR2087588
  3. [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). Zbl1056.37006MR2015618
  4. [4] F. Balibrea & J. Smítal – « Strong distributional chaos and minimal sets », Topology Appl.156 (2009), p. 1673–1678. Zbl1175.37034MR2521703
  5. [5] F. Balibrea, J. Smítal & M. Štefánková – « The three versions of distributional chaos », Chaos Solitons Fractals23 (2005), p. 1581–1583. Zbl1069.37013MR2101573
  6. [6] J. Banks – « Regular periodic decompositions for topologically transitive maps », Ergodic Theory Dynam. Systems17 (1997), p. 505–529. Zbl0921.54029MR1452178
  7. [7] F. Blanchard, B. Host & S. Ruette – « Asymptotic pairs in positive-entropy systems », Ergodic Theory Dynam. Systems22 (2002), p. 671–686. Zbl1018.37005MR1908549
  8. [8] F. Blanchard & J. Kwiatkowski – « Minimal self-joinings and positive topological entropy. II », Studia Math.128 (1998), p. 121–133. Zbl0909.54034MR1490816
  9. [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. Zbl1096.37002MR2180227
  10. [10] H. Furstenberg – « Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation », Math. Systems Theory1 (1967), p. 1–49. Zbl0146.28502MR213508
  11. [11] —, Recurrence in ergodic theory and combinatorial number theory, Princeton Univ. Press, 1981. Zbl0459.28023MR603625
  12. [12] C. Grillenberger – « Constructions of strictly ergodic systems. I. Given entropy », Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/73), p. 323–334. Zbl0253.28004MR340544
  13. [13] —, « Constructions of strictly ergodic systems. II. K -Systems », Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/73), p. 335–342. Zbl0253.28005MR340545
  14. [14] C. Grillenberger & P. Shields – « Construction of strictly ergodic systems. III. Bernoulli systems », Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33 (1975/76), p. 215–217. Zbl0348.60046MR396908
  15. [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. Zbl0944.28015MR1713534
  16. [16] W. Huang & X. Ye – « Devaney’s chaos and 2-scattering imply Li-Yorke’s chaos », Topology117 (2002), p. 259–272. Zbl0997.54061MR1874089
  17. [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. Zbl1103.37002MR2180232
  18. [18] W. Huang & X. Ye – « A local variational relation and applications », Israel J. Math.151 (2006), p. 237–279. Zbl1122.37013MR2214126
  19. [19] D. Kerr & H. Li – « Independence in topological and C * -dynamics », Math. Ann.338 (2007), p. 869–926. Zbl1131.46046MR2317754
  20. [20] P. Kurka – Topological and symbolic dynamics, Cours Spécialisés, vol. 11, Soc. Math. France, 2003. Zbl1038.37011MR2041676
  21. [21] T. Y. Li & J. A. Yorke – « Period three implies chaos », Amer. Math. Monthly82 (1975), p. 985–992. Zbl0351.92021MR385028
  22. [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. Zbl1010.37019MR2021982
  23. [23] P. Oprocha – « Distributional chaos revisited », Trans. Amer. Math. Soc.361 (2009), p. 4901–4925. Zbl1179.37017MR2506431
  24. [24] —, « Weak mixing and product recurrence », Ann. Inst. Fourier60 (2010), p. 1233–1257. Zbl1203.37026MR2722240
  25. [25] P. Oprocha & M. Štefánková – « Specification property and distributional chaos almost everywhere », Proc. Amer. Math. Soc.136 (2008), p. 3931–3940. Zbl1159.37004MR2425733
  26. [26] P. Oprocha & P. Wilczyński – « Distributional chaos via semiconjugacy », Nonlinearity20 (2007), p. 2661–2679. Zbl1131.37017MR2361250
  27. [27] R. Pikuła – « On some notions of chaos in dimension zero », Colloq. Math.107 (2007), p. 167–177. Zbl1130.37327MR2284159
  28. [28] B. Schweizer, A. Sklar & J. Smítal – « Distributional (and other) chaos and its measurement », Real Anal. Exchange 26 (2000/01), p. 495–524. Zbl1012.37022MR1844132
  29. [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. Zbl0812.58062MR1227094
  30. [30] A. Sklar & J. Smítal – « Distributional chaos on compact metric spaces via specification properties », J. Math. Anal. Appl.241 (2000), p. 181–188. Zbl1060.37012MR1739200
  31. [31] F. Tan & J. Xiong – « Chaos via Furstenberg family couple », Topology Appl.156 (2009), p. 525–532. Zbl1161.37019MR2492300
  32. [32] S. Williams – « Toeplitz minimal flows which are not uniquely ergodic », Z. Wahrsch. Verw. Gebiete67 (1984), p. 95–107. Zbl0584.28007MR756807

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