×
Oprocha, Piotr

Distributional chaos revisited. (English) Zbl 1179.37017

Trans. Am. Math. Soc. 361, No. 9, 4901-4925 (2009).
The modification of the definition of distributionally scrambled set in a way that it becomes significant from the topological point of view, that is, to avoid its concentration on small sets is the main idea of this work. In an example it shown that the definition of distributional chaos may be fulfilled by a dynamical system with regular dynamics embedded.
The basic results are stated in 35 theorems.
Reviewer: Gasanbek T. Arazov (Baku)

Cited in 49 Documents

MSC:

37B10 Symbolic dynamics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Keywords:

distributional chaos; chaotic pair; Li-Yorke chaos
PDF BibTeX XML Cite
\textit{P. Oprocha}, Trans. Am. Math. Soc. 361, No. 9, 4901--4925 (2009; Zbl 1179.37017)
Full Text: DOI

References:

[1] Ethan Akin and Sergiĭ Kolyada, Li-Yorke sensitivity, Nonlinearity 16 (2003), no. 4, 1421 – 1433. · Zbl 1045.37004
[2] Lluís Alsedà, Jaume Llibre, and Michał Misiurewicz, Combinatorial dynamics and entropy in dimension one, Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. · Zbl 0963.37001
[3] Joseph Auslander, Minimal flows and their extensions, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988. Notas de Matemática [Mathematical Notes], 122. · Zbl 0654.54027
[4] Joseph Auslander and James A. Yorke, Interval maps, factors of maps, and chaos, Tôhoku Math. J. (2) 32 (1980), no. 2, 177 – 188. · Zbl 0448.54040
[5] Marta Babilonová-Štefánková, Extreme chaos and transitivity, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), no. 7, 1695 – 1700. Dynamical systems and functional equations (Murcia, 2000). · Zbl 1056.37042
[6] F. Balibrea, L. Reich, and J. Smítal, Iteration theory: dynamical systems and functional equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), no. 7, 1627 – 1647. Dynamical systems and functional equations (Murcia, 2000). · Zbl 1056.37003
[7] F. Balibrea, B. Schweizer, A. Sklar, and J. Smítal, Generalized specification property and distributional chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), no. 7, 1683 – 1694. Dynamical systems and functional equations (Murcia, 2000). · Zbl 1056.37006
[8] F. Balibrea, J. Smítal, and M. Štefánková, The three versions of distributional chaos, Chaos Solitons Fractals 23 (2005), no. 5, 1581 – 1583. · Zbl 1069.37013
[9] John Banks, Topological mapping properties defined by digraphs, Discrete Contin. Dynam. Systems 5 (1999), no. 1, 83 – 92. · Zbl 0957.54020
[10] François Blanchard, Eli Glasner, Sergiĭ Kolyada, and Alejandro Maass, On Li-Yorke pairs, J. Reine Angew. Math. 547 (2002), 51 – 68. · Zbl 1059.37006
[11] François Blanchard, Wen Huang, and L’ubomír Snoha, Topological size of scrambled sets, Colloq. Math. 110 (2008), no. 2, 293 – 361. · Zbl 1146.37008
[12] Louis Block, Homoclinic points of mappings of the interval, Proc. Amer. Math. Soc. 72 (1978), no. 3, 576 – 580. · Zbl 0365.58015
[13] Rufus Bowen, Topological entropy and axiom \?, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 23 – 41.
[14] A. M. Bruckner and Thakyin Hu, On scrambled sets for chaotic functions, Trans. Amer. Math. Soc. 301 (1987), no. 1, 289 – 297. · Zbl 0639.26004
[15] Lidong Wang, Zhizhi Chen, and Gongfu Liao, The complexity of a minimal sub-shift on symbolic spaces, J. Math. Anal. Appl. 317 (2006), no. 1, 136 – 145. · Zbl 1086.37006
[16] Lidong Wang, Gongfu Liao, Zhenyan Chu, and Xiaodong Duan, The set of recurrent points of a continuous self-map on an interval and strong chaos, J. Appl. Math. Comput. 14 (2004), no. 1-2, 277 – 288. · Zbl 1043.37032
[17] Jesse Paul Clay, Proximity relations in transformation groups, Trans. Amer. Math. Soc. 108 (1963), 88 – 96. · Zbl 0115.40301
[18] W. A. Coppel, Maps of an interval, IMA Preprint Series, vol. 26. · Zbl 0606.54032
[19] Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. · Zbl 0328.28008
[20] Robert L. Devaney, A first course in chaotic dynamical systems, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1992. Theory and experiment; With a separately available computer disk. · Zbl 0768.58001
[21] Bau-Sen Du, On the invariance of Li-Yorke chaos of interval maps, J. Difference Equ. Appl. 11 (2005), no. 9, 823 – 828. · Zbl 1076.37024
[22] V. V. Fedorenko, A. N. Šarkovskii, and J. Smítal, Characterizations of weakly chaotic maps of the interval, Proc. Amer. Math. Soc. 110 (1990), no. 1, 141 – 148. · Zbl 0728.26008
[23] T. Gedeon, There are no chaotic mappings with residual scrambled sets, Bull. Austral. Math. Soc. 36 (1987), no. 3, 411 – 416. · Zbl 0646.26008
[24] Eli Glasner and Benjamin Weiss, Sensitive dependence on initial conditions, Nonlinearity 6 (1993), no. 6, 1067 – 1075. · Zbl 0790.58025
[25] Shmuel Glasner, Compressibility properties in topological dynamics, Amer. J. Math. 97 (1975), 148 – 171. · Zbl 0298.54023
[26] Shmuel Glasner, Proximal flows, Lecture Notes in Mathematics, Vol. 517, Springer-Verlag, Berlin-New York, 1976. · Zbl 0322.54017
[27] Wen Huang and Xiangdong Ye, Homeomorphisms with the whole compacta being scrambled sets, Ergodic Theory Dynam. Systems 21 (2001), no. 1, 77 – 91. · Zbl 0978.37003
[28] A. Iwanik, Independence and scrambled sets for chaotic mappings, The mathematical heritage of C. F. Gauss, World Sci. Publ., River Edge, NJ, 1991, pp. 372 – 378. · Zbl 0780.58015
[29] Wen Huang and Xiangdong Ye, Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos, Topology Appl. 117 (2002), no. 3, 259 – 272. · Zbl 0997.54061
[30] K. Janková and J. Smítal, A characterization of chaos, Bull. Austral. Math. Soc. 34 (1986), no. 2, 283 – 292. · Zbl 0577.54041
[31] I. Kan, A chaotic function possessing a scrambled set with positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), no. 1, 45 – 49. , https://doi.org/10.1090/S0002-9939-1984-0749887-4 J. Smítal, A chaotic function with a scrambled set of positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), no. 1, 50 – 54. · Zbl 0592.26005
[32] S. F. Kolyada, Li-Yorke sensitivity and other concepts of chaos, Ukraïn. Mat. Zh. 56 (2004), no. 8, 1043 – 1061 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 56 (2004), no. 8, 1242 – 1257. · Zbl 1075.37500
[33] Shi Hai Li, \?-chaos and topological entropy, Trans. Amer. Math. Soc. 339 (1993), no. 1, 243 – 249. · Zbl 0812.54046
[34] T. Y. Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985 – 992. · Zbl 0351.92021
[35] Gongfu Liao and Lidong Wang, Almost periodicity and distributional chaos, Foundations of computational mathematics (Hong Kong, 2000) World Sci. Publ., River Edge, NJ, 2002, pp. 189 – 210. · Zbl 1010.37019
[36] Jaume Llibre and Michał Misiurewicz, Horseshoes, entropy and periods for graph maps, Topology 32 (1993), no. 3, 649 – 664. · Zbl 0787.54021
[37] Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8, Springer-Verlag, Berlin, 1987. Translated from the Portuguese by Silvio Levy. · Zbl 0616.28007
[38] Mario Martelli, Mai Dang, and Tanya Seph, Defining chaos, Math. Mag. 71 (1998), no. 2, 112 – 122. · Zbl 1008.37014
[39] Michał Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 2, 167 – 169 (English, with Russian summary). · Zbl 0459.54031
[40] Michał Misiurewicz, Chaos almost everywhere, Iteration theory and its functional equations (Lochau, 1984) Lecture Notes in Math., vol. 1163, Springer, Berlin, 1985, pp. 125 – 130. · Zbl 0625.58007
[41] Marston Morse and Gustav A. Hedlund, Symbolic Dynamics, Amer. J. Math. 60 (1938), no. 4, 815 – 866. · Zbl 0019.33502
[42] Marston Morse and Gustav A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62 (1940), 1 – 42. · Zbl 0022.34003
[43] Elena Murinová, Generic chaos in metric spaces, Acta Univ. M. Belii Ser. Math. 8 (2000), 43 – 50. · Zbl 1183.37018
[44] Piotr Oprocha, Relations between distributional and Devaney chaos, Chaos 16 (2006), no. 3, 033112, 5. · Zbl 1146.37301
[45] Piotr Oprocha, Specification properties and dense distributional chaos, Discrete Contin. Dyn. Syst. 17 (2007), no. 4, 821 – 833. · Zbl 1120.37009
[46] Piotr Oprocha and Paweł Wilczyński, Shift spaces and distributional chaos, Chaos Solitons Fractals 31 (2007), no. 2, 347 – 355. · Zbl 1140.37303
[47] Rafał Pikuła, On some notions of chaos in dimension zero, Colloq. Math. 107 (2007), no. 2, 167 – 177. · Zbl 1130.37327
[48] Józef Piórek, On the generic chaos in dynamical systems, Univ. Iagel. Acta Math. 25 (1985), 293 – 298. · Zbl 0587.54061
[49] David Ruelle and Floris Takens, On the nature of turbulence, Comm. Math. Phys. 20 (1971), 167 – 192. · Zbl 0223.76041
[50] Sylvie Ruette, Dense chaos for continuous interval maps, Nonlinearity 18 (2005), no. 4, 1691 – 1698. · Zbl 1181.37056
[51] B. Schweizer, A. Sklar, and J. Smítal, Distributional (and other) chaos and its measurement, Real Anal. Exchange 26 (2000/01), no. 2, 495 – 524. · Zbl 1012.37022
[52] B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (1994), no. 2, 737 – 754. · Zbl 0812.58062
[53] A. Sklar and J. Smítal, Distributional chaos on compact metric spaces via specification properties, J. Math. Anal. Appl. 241 (2000), no. 2, 181 – 188. · Zbl 1060.37012
[54] J. Smítal, A chaotic function with some extremal properties, Proc. Amer. Math. Soc. 87 (1983), no. 1, 54 – 56. · Zbl 0555.26003
[55] I. Kan, A chaotic function possessing a scrambled set with positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), no. 1, 45 – 49. , https://doi.org/10.1090/S0002-9939-1984-0749887-4 J. Smítal, A chaotic function with a scrambled set of positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), no. 1, 50 – 54. · Zbl 0592.26005
[56] J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), no. 1, 269 – 282. · Zbl 0639.54029
[57] J. Smítal, Various notions of chaos, recent results, open problems, Real Anal. Exchange 26th Summer Symposium Conference, suppl. (2002), 81 – 85. Report on the Summer Symposium in Real Analysis XXVI. · Zbl 1192.37016
[58] Jaroslav Smítal and Marta Štefánková, Distributional chaos for triangular maps, Chaos Solitons Fractals 21 (2004), no. 5, 1125 – 1128. · Zbl 1060.37037
[59] Ľubomír Snoha, Generic chaos, Comment. Math. Univ. Carolin. 31 (1990), no. 4, 793 – 810. · Zbl 0724.58044
[60] Ľubomír Snoha, Dense chaos, Comment. Math. Univ. Carolin. 33 (1992), no. 4, 747 – 752. · Zbl 0784.58043
[61] Stephen Wiggins, Chaotic transport in dynamical systems, Interdisciplinary Applied Mathematics, vol. 2, Springer-Verlag, New York, 1992. · Zbl 0747.34028
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.