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Recent development of chaos theory in topological dynamics. (English) Zbl 1335.54039

Summary: We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.

MSC:

54H20 Topological dynamics (MSC2010)
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B40 Topological entropy
54-02 Research exposition (monographs, survey articles) pertaining to general topology
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
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[1] Adler, R. L., Konheim, A. G., McAndrew, M. H.: Topological entropy. Trans. Amer. Math. Soc., 114, 309-319 (1965) · Zbl 0127.13102
[2] Akin, E.: Recurrence in topological dynamics: Furstenberg families and Ellis actions. The University Series in Mathematics, Plenum Press, New York, 1997 · Zbl 0919.54033
[3] Akin, E.; Auslander, J.; Berg, K., When is a transitive map chaotic?, 25-40 (1996) · Zbl 0861.54034
[4] Akin, E., Glasner, E.: Residual properties and almost equicontinuity. J. Anal. Math., 84, 243-286 (2001) · Zbl 1182.37009
[5] Akin, E., Glasner, E., Huang, W., et al.: Sufficient conditions under which a transitive system is chaotic. Ergodic Theory Dynam. Systems, 30(5), 1277-1310 (2010) · Zbl 1211.37001
[6] Akin, E., Kolyada, S.: Li-Yorke sensitivity. Nonlinearity, 16(4), 1421-1433 (2003) · Zbl 1045.37004
[7] Auslander, J., Minimal flows and their extensions (1988), Amsterdam · Zbl 0654.54027
[8] Auslander, J., Yorke, J. A.: Interval maps, factors of maps, and chaos. Tohoku Math. J. (2), 32(2), 177-188 (1980) · Zbl 0448.54040
[9] Balibrea, F., Guirao, J. L. G., Oprocha, P.: On invariant-scrambled sets. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20(9), 2925-2935 (2010) · Zbl 1202.37016
[10] Balibrea, F., Smítal, J., Stefánková, M.: The three versions of distributional chaos. Chaos Solitons Fractals, 23(5), 1581-1583 (2005) · Zbl 1069.37013
[11] Banks, J.: Chaos for induced hyperspace maps. Chaos Solitons Fractals, 25(3), 681-685 (2005) · Zbl 1071.37012
[12] Banks, J., Brooks, J., Cairns, G., et al.: On Devaney’s definition of chaos. Amer. Math. Monthly, 99(4), 332-334 (1992) · Zbl 0758.58019
[13] Bauer, W., Sigmund, K.: Topological dynamics of transformations induced on the space of probability measures. Monatsh. Math., 79, 81-92 (1975) · Zbl 0314.54042
[14] Blanchard, F.: Topological chaos: what may this mean? J. Difference Equ. Appl., 15(1), 23-46 (2009) · Zbl 1253.37013
[15] Blanchard, F., Glasner, E., Kolyada, S., et al.: On Li-Yorke pairs. J. Reine Angew. Math., 547, 51-68 (2002) · Zbl 1059.37006
[16] Blanchard, F., Host, B., Ruette, S.: Asymptotic pairs in positive-entropy systems. Ergodic Theory Dynam. Systems, 22(3), 671-686 (2002) · Zbl 1018.37005
[17] Blanchard, F., Huang, W.: Entropy sets, weakly mixing sets and entropy capacity. Discrete Contin. Dyn. Syst., 20(2), 275-311 (2008) · Zbl 1151.37019
[18] Blanchard, F., Huang, W., Snoha, L.: Topological size of scrambled sets. Colloq. Math., 110(2), 293-361 (2008) · Zbl 1146.37008
[19] Devaney, R. L.: An Introduction to Chaotic Dynamical Systems. Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, second edition, 1989 · Zbl 0695.58002
[20] Dolezelová, J.: Scrambled and distributionally scrambled n-tuples. J. Difference Equ. Appl., 20(8), 1169-1177 (2014) · Zbl 1321.37030
[21] Downarowicz, T.: Positive topological entropy implies chaos DC2. Proc. Amer. Math. Soc., 142(1), 137-149 (2014) · Zbl 1304.37008
[22] Du, B.-S.: On the invariance of Li-Yorke chaos of interval maps. J. Difference Equ. Appl., 11(9), 823-828 (2005) · Zbl 1076.37024
[23] Fomin, S.: On dynamical systems with a purely point spectrum. Doklady Akad. Nauk SSSR (N.S.), 77, 29-32 (1951) · Zbl 0045.38803
[24] Forys, M.; Huang, W.; Li, J.; etal., Invariant scrambled sets, uniform rigidity and weak mixing (2014)
[25] Furstenberg, H.: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory, 1, 1-49 (1967) · Zbl 0146.28502
[26] Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981 · Zbl 0459.28023
[27] García-Ramos, F.: Weak forms of topological and measure theoretical equicontinuity: relationships with discrete spectrum and sequence entropy. Preprint, arXiv:1402.7327[math.DS] · Zbl 1366.37019
[28] Glasner, E., Ergodic theory via joinings (2003) · Zbl 1038.37002
[29] Glasner, E., Weiss, B.: Sensitive dependence on initial conditions. Nonlinearity, 6(6), 1067-1075 (1993) · Zbl 0790.58025
[30] Glasner, E., Weiss, B.: Quasi-factors of zero-entropy systems. J. Amer. Math. Soc., 8(3), 665-686 (1995) · Zbl 0846.28009
[31] Glasner, E., Ye, X.: Local entropy theory. Ergodic Theory Dynam. Systems, 29(2), 321-356 (2009) · Zbl 1160.37309
[32] Glasner, S.: Proximal flows. In: Lecture Notes in Mathematics, Vol. 517, Springer-Verlag, Berlin-New York, 1976 · Zbl 0322.54017
[33] Glasner, S., Maon, D.: Rigidity in topological dynamics. Ergodic Theory Dynam. Systems, 9(2), 309-320 (1989) · Zbl 0661.58027
[34] Gottschalk, W. H., Hedlund, G. A.: Topological dynamics. In: American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R. I., 1955 · Zbl 0067.15204
[35] Guckenheimer, J.: Sensitive dependence to initial conditions for one-dimensional maps. Comm. Math. Phys., 70(2), 133-160 (1979) · Zbl 0429.58012
[36] Guirao, J. L. G., Kwietniak, D., Lampart, M., et al.: Chaos on hyperspaces. Nonlinear Anal., 71(1-2), 1-8 (2009) · Zbl 1175.37024
[37] Huang, W.: Stable sets and -stable sets in positive-entropy systems. Comm. Math. Phys., 279(2), 535-557 (2008) · Zbl 1167.37013
[38] Huang, W.; Kolyada, S.; Zhang, G., Multi-sensitivity, Lyapunov numbers and almost automorphic maps (2014)
[39] Huang, W., Li, H., Ye, X.: Family independence for topological and measurable dynamics. Trans. Amer. Math. Soc., 364(10), 5209-5242 (2012) · Zbl 1286.37017
[40] Huang, W., Li, J., Ye, X.: Stable sets and mean Li-Yorke chaos in positive entropy systems. J. Funct. Anal., 266(6), 3377-3394 (2014) · Zbl 1364.37016
[41] Huang, W.; Li, J.; Ye, X.; Zhou, X., Topological entropy and diagonal-weakly mixing sets (2014)
[42] Huang, W., Lu, P., Ye, X.: Measure-theoretical sensitivity and equicontinuity. Israel J. Math., 183, 233-283 (2011) · Zbl 1257.37018
[43] Huang, W., Shao, S., Ye, X.: Mixing and proximal cells along sequences. Nonlinearity, 17(4), 1245-1260 (2004) · Zbl 1055.37014
[44] Huang, W., Xu, L., Yi, Y.: Asymptotic pairs, stable sets and chaos in positive entropy systems. J. Funct. Anal., to appear · Zbl 1317.37014
[45] Huang, W., Ye, X.: Homeomorphisms with the whole compacta being scrambled sets. Ergodic Theory Dynam. Systems, 21(1), 77-91 (2001) · Zbl 0978.37003
[46] Huang, W., Ye, X.: Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos. Topology Appl., 117(3), 259-272 (2002) · Zbl 0997.54061
[47] Huang, W., Ye, X.: Topological complexity, return times and weak disjointness. Ergodic Theory Dynam. Systems, 24(3), 825-846 (2004) · Zbl 1052.37005
[48] Huang, W., Ye, X.: Dynamical systems disjoint from any minimal system. Trans. Amer. Math. Soc., 357(2), 669-694 (2005) · Zbl 1072.37011
[49] Huang, W., Ye, X.: A local variational relation and applications. Israel J. Math., 151, 237-279 (2006) · Zbl 1122.37013
[50] Iwanik, A., Independence and scrambled sets for chaotic mappings, 372-378 (1991), NJ · Zbl 0780.58015
[51] Janková, K., Smítal, J.: A characterization of chaos. Bull. Austral. Math. Soc., 34(2), 283-292 (1986) · Zbl 0577.54041
[52] Katznelson, Y.; Weiss, B., When all points are recurrent/generic, 195-210 (1981) · Zbl 0469.54023
[53] Kerr, D., Li, H.: Independence in topological and C*-dynamics. Math. Ann., 338(4), 869-926 (2007) · Zbl 1131.46046
[54] Kerr, D., Li, H.: Combinatorial independence and sofic entropy. Commun. Math. Stat., 1(2), 213-257 (2013) · Zbl 1315.37025
[55] Keynes, H. B., Robertson, J. B.: Eigenvalue theorems in topological transformation groups. Trans. Amer. Math. Soc., 139, 359-369 (1969) · Zbl 0176.20602
[56] Kolyada, S.: Li-Yorke sensitivity and other concepts of chaos. Ukrain. Mat. Zh., 56(8), 1043-1061 (2004) · Zbl 1075.37500
[57] Kolyada, S.; Snoha, L., Some aspects of topological transitivity — a survey, 3-35 (1997), Graz · Zbl 0907.54036
[58] Li, J.: Chaos and entropy for interval maps. J. Dynam. Differential Equations, 23(2), 333-352 (2011) · Zbl 1217.37041
[59] Li, J.: Transitive points via Furstenberg family. Topology Appl., 158(16), 2221-2231 (2011) · Zbl 1234.37034
[60] Li, J.: Equivalent conditions of devaney chaos on the hyperspace. J. Univ. Sci. Technol. China, 44(2), 93-95 (2014)
[61] Li, J.: Localization of mixing property via Furstenberg families. Discrete Contin. Dyn. Syst., 35(2), 725-740 (2015) · Zbl 1310.54026
[62] Li, J., Oprocha, P.: On n-scrambled tuples and distributional chaos in a sequence. J. Difference Equ. Appl., 19(6), 927-941 (2013) · Zbl 1315.54029
[63] Li, J.; Oprocha, P.; Zhang, G., On recurrence over subsets and weak mixing (2013)
[64] Li, J., Tu, S.: On proximality with Banach density one. J. Math. Anal. Appl., 416(1), 36-51 (2014) · Zbl 1317.37026
[65] Li, J., Tu, S., Ye, X.: Mean equicontinuity and mean sensitivity. Ergodic Theory Dynam. Systems, to appear · Zbl 1356.37016
[66] Li, J., Yan, K., Ye, X.: Recurrence properties and disjointness on the induced spaces. Discrete Contin. Dyn. Syst., 35(3), 1059-1073 (2015) · Zbl 1310.54027
[67] Li, R.; Shi, Y., Stronger forms of sensitivity for measure-preserving maps and semiflows on probability spaces, 10 (2014) · Zbl 1474.37002
[68] Li, S.: ω-chaos and topological entropy. Trans. Amer. Math. Soc., 339(1), 243-249 (1993) · Zbl 0812.54046
[69] Li, T., Yorke, J. A.: Period three implies chaos. Amer. Math. Monthly, 82(10), 985-992 (1975) · Zbl 0351.92021
[70] Liao, G., Fan, Q.: Minimal subshifts which display Schweizer-Smítal chaos and have zero topological entropy. Sci. China Ser. A, 41(1), 33-38 (1998) · Zbl 0931.54034
[71] Liu, H.; Liao, L.; Wang, L., Thickly syndetical sensitivity of topological dynamical system, 4 (2014) · Zbl 1419.37010
[72] Lorenz, E. N.: Deterministic nonperiodic flow. J. Atmospheric Sci., 20(2), 130-148 (1963) · Zbl 1417.37129
[73] Mai, J.: Continuous maps with the whole space being a scrambled set. Chinese Sci. Bull., 42(19), 1603-1606 (1997) · Zbl 0930.37010
[74] Mai, J.: The structure of equicontinuous maps. Trans. Amer. Math. Soc., 355(10), 4125-4136 (2003) · Zbl 1026.54031
[75] Mai, J.: Devaney’s chaos implies existence of s-scrambled sets. Proc. Amer. Math. Soc., 132(9), 2761-2767 (2004) · Zbl 1055.54019
[76] Moothathu, T. K. S.: Stronger forms of sensitivity for dynamical systems. Nonlinearity, 20(9), 2115-2126 (2007) · Zbl 1132.54023
[77] Moothathu, T. K. S.: Syndetically proximal pairs. J. Math. Anal. Appl., 379(2), 656-663 (2011) · Zbl 1225.54011
[78] Moothathu, T. K. S., Oprocha, P.: Syndetic proximality and scrambled sets. Topol. Methods Nonlinear Anal., 41(2), 421-461 (2013) · Zbl 1327.37006
[79] Mycielski, J.: Independent sets in topological algebras. Fund. Math., 55, 139-147 (1964) · Zbl 0124.01301
[80] Oprocha, P.: Relations between distributional and Devaney chaos. Chaos, 16(3), 033112, 5 (2006) · Zbl 1146.37301
[81] Oprocha, P.: Minimal systems and distributionally scrambled sets. Bull. Soc. Math. France, 140(3), 401-439 (2012) · Zbl 1278.37013
[82] Oprocha, P., Zhang, G.: On local aspects of topological weak mixing in dimension one and beyond. Studia Math., 202(3), 261-288 (2011) · Zbl 1217.37012
[83] Oprocha, P., Zhang, G.: On weak product recurrence and synchronization of return times. Adv. Math., 244, 395-412 (2013) · Zbl 1286.37015
[84] Oprocha, P., Zhang, G.: On local aspects of topological weak mixing, sequence entropy and chaos. Ergodic Theory Dynam. Systems, 34(5), 1615-1639 (2014) · Zbl 1322.37003
[85] Oprocha, P.; Zhang, G., Topological aspects of dynamics of pairs, tuples and sets, 665-709 (2014), Paris · Zbl 1305.54002
[86] Pikula, R.: On some notions of chaos in dimension zero. Colloq. Math., 107(2), 167-177 (2007) · Zbl 1130.37327
[87] Román-Flores, H.: A note on transitivity in set-valued discrete systems. Chaos Solitons Fractals, 17(1), 99-104 (2003) · Zbl 1098.37008
[88] Ruelle, D., Dynamical systems with turbulent behavior, 341-360 (1978), Berlin
[89] Ruelle, D., Takens, F.: On the nature of turbulence. Comm. Math. Phys., 20, 167-192 (1971) · Zbl 0223.76041
[90] Ruette, S.: Transitive sensitive subsystems for interval maps. Studia Math., 169(1), 81-104 (2005) · Zbl 1094.37020
[91] Scarpellini, B.: Stability properties of flows with pure point spectrum. J. London Math. Soc. (2), 26(3), 451-464 (1982) · Zbl 0505.28011
[92] 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
[93] Shao, S.: Proximity and distality via Furstenberg families. Topology Appl., 153(12), 2055-2072 (2006) · Zbl 1099.37006
[94] Shao, S., Ye, X., Zhang, R.: Sensitivity and regionally proximal relation in minimal systems. Sci. China Ser. A, 51(6), 987-994 (2008) · Zbl 1155.37305
[95] Smítal, J.: A chaotic function with some extremal properties. Proc. Amer. Math. Soc., 87(1), 54-56 (1983) · Zbl 0555.26003
[96] Smítal, J.: Chaotic functions with zero topological entropy. Trans. Amer. Math. Soc., 297(1), 269-282 (1986) · Zbl 0639.54029
[97] Smítal, J., Topological entropy and distributional chaos, 61-65 (2006) · Zbl 1117.37009
[98] Smítal, J., Stefánková, M.: Distributional chaos for triangular maps. Chaos Solitons Fractals, 21(5), 1125-1128 (2004) · Zbl 1060.37037
[99] Snoha, L.: Generic chaos. Comment. Math. Univ. Carolin., 31(4), 793-810 (1990) · Zbl 0724.58044
[100] Tan, F., Fu, H.: On distributional n-chaos. Acta Math. Sci. Ser. B Engl. Ed., 34(5), 1473-1480 (2014) · Zbl 1324.37007
[101] Tan, F., Xiong, J.: Chaos via Furstenberg family couple. Topology Appl., 156(3), 525-532 (2009) · Zbl 1161.37019
[102] Tan, F., Zhang, R.: On F-sensitive pairs. Acta Math. Sci. Ser. B Engl. Ed., 31(4), 1425-1435 (2011) · Zbl 1249.54076
[103] Walters, P., An introduction to ergodic theory (1982), New York · Zbl 0475.28009
[104] Wang, H.; Xiong, J.; Tan, F., Furstenberg families and sensitivity, 12 (2010) · Zbl 1187.37016
[105] Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos (1990), New York · Zbl 0701.58001
[106] Xiong, J.: A chaotic map with topological entropy. Acta Math. Sci. English Ed., 6(4), 439-443 (1986) · Zbl 0659.58038
[107] Xiong, J.: Chaos in a topologically transitive system. Sci. China Ser. A, 48(7), 929-939 (2005) · Zbl 1096.37018
[108] Xiong, J., LÜ, J., Tan, F.: Furstenberg family and chaos. Sci. China Ser. A, 50(9), 1325-1333 (2007) · Zbl 1136.54025
[109] Xiong, J., Yang, Z.: Chaos caused by a topologically mixing map. In: Dynamical systems and related topics (Nagoya, 1990), Adv. Ser. Dynam. Systems, Vol. 9, 550-572, World Sci. Publ., River Edge, NJ, 1991
[110] Ye, X.; Yu, T., Sensitivity, proximal extension and higher order almost automorphy (2014)
[111] Ye, X., Zhang, R.: On sensitive sets in topological dynamics. Nonlinearity, 21(7), 1601-1620 (2008) · Zbl 1153.37322
[112] Yuan, D., LÜ, J.: Invariant scrambled sets in transitive systems. Adv. Math. (China), 38(3), 302-308 (2009) · Zbl 1482.37014
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