Distributional chaos revisited

Oprocha, Piotr

doi:10.1090/S0002-9947-09-04810-7

Distributional chaos revisited
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Trans. Amer. Math. Soc. 361 (2009), 4901-4925 Request permission

Abstract:

In their famous paper, Schweizer and Smítal introduced the definition of a distributionally chaotic pair and proved that the existence of such a pair implies positive topological entropy for continuous mappings of a compact interval. Further, their approach was extended to the general compact metric space case.

In this article we provide an example which shows that the definition of distributional chaos (and as a result Li-Yorke chaos) may be fulfilled by a dynamical system with (intuitively) regular dynamics embedded in $\mathbb {R}^3$. Next, we state strengthened versions of distributional chaos which, as we show, are present in systems commonly considered to have complex dynamics.

We also prove that any interval map with positive topological entropy contains two invariant subsets $X,Y \subset I$ such that $f|_X$ has positive topological entropy and $f|_Y$ displays distributional chaos of type $1$, but not conversely.

References

Ethan Akin and Sergiĭ Kolyada, Li-Yorke sensitivity, Nonlinearity 16 (2003), no. 4, 1421–1433. MR 1986303, DOI 10.1088/0951-7715/16/4/313
Lluís Alsedà, Jaume Llibre, and MichałMisiurewicz, Combinatorial dynamics and entropy in dimension one, Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1255515, DOI 10.1142/1980
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. MR 956049
Joseph Auslander and James A. Yorke, Interval maps, factors of maps, and chaos, Tohoku Math. J. (2) 32 (1980), no. 2, 177–188. MR 580273, DOI 10.2748/tmj/1178229634
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). MR 2015619, DOI 10.1142/S0218127403007540
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). MR 2015613, DOI 10.1142/S0218127403007485
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). MR 2015618, DOI 10.1142/S0218127403007539
F. Balibrea, J. Smítal, and M. Štefánková, The three versions of distributional chaos, Chaos Solitons Fractals 23 (2005), no. 5, 1581–1583. MR 2101573, DOI 10.1016/j.chaos.2004.06.011
John Banks, Topological mapping properties defined by digraphs, Discrete Contin. Dynam. Systems 5 (1999), no. 1, 83–92. MR 1664461, DOI 10.3934/dcds.1999.5.83
François Blanchard, Eli Glasner, Sergiĭ Kolyada, and Alejandro Maass, On Li-Yorke pairs, J. Reine Angew. Math. 547 (2002), 51–68. MR 1900136, DOI 10.1515/crll.2002.053
François Blanchard, Wen Huang, and L’ubomír Snoha, Topological size of scrambled sets, Colloq. Math. 110 (2008), no. 2, 293–361. MR 2353910, DOI 10.4064/cm110-2-3
Louis Block, Homoclinic points of mappings of the interval, Proc. Amer. Math. Soc. 72 (1978), no. 3, 576–580. MR 509258, DOI 10.1090/S0002-9939-1978-0509258-X
Rufus Bowen, Topological entropy and axiom $\textrm {A}$, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 23–41. MR 0262459
A. M. Bruckner and Thakyin Hu, On scrambled sets for chaotic functions, Trans. Amer. Math. Soc. 301 (1987), no. 1, 289–297. MR 879574, DOI 10.1090/S0002-9947-1987-0879574-0
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. MR 2205317, DOI 10.1016/j.jmaa.2005.12.069
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. MR 2025438, DOI 10.1007/BF02936114
Jesse Paul Clay, Proximity relations in transformation groups, Trans. Amer. Math. Soc. 108 (1963), 88–96. MR 154269, DOI 10.1090/S0002-9947-1963-0154269-3
W. A. Coppel, Maps of an interval, IMA Preprint Series, vol. 26.
Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675, DOI 10.1007/BFb0082364
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. MR 1202237
Bau-Sen Du, On the invariance of Li-Yorke chaos of interval maps, J. Difference Equ. Appl. 11 (2005), no. 9, 823–828. MR 2159799, DOI 10.1080/10236190500138320
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. MR 1017846, DOI 10.1090/S0002-9939-1990-1017846-5
T. Gedeon, There are no chaotic mappings with residual scrambled sets, Bull. Austral. Math. Soc. 36 (1987), no. 3, 411–416. MR 923822, DOI 10.1017/S0004972700003695
Eli Glasner and Benjamin Weiss, Sensitive dependence on initial conditions, Nonlinearity 6 (1993), no. 6, 1067–1075. MR 1251259, DOI 10.1088/0951-7715/6/6/014
Shmuel Glasner, Compressibility properties in topological dynamics, Amer. J. Math. 97 (1975), 148–171. MR 365537, DOI 10.2307/2373665
Shmuel Glasner, Proximal flows, Lecture Notes in Mathematics, Vol. 517, Springer-Verlag, Berlin-New York, 1976. MR 0474243, DOI 10.1007/BFb0080139
Wen Huang and Xiangdong Ye, Homeomorphisms with the whole compacta being scrambled sets, Ergodic Theory Dynam. Systems 21 (2001), no. 1, 77–91. MR 1826661, DOI 10.1017/S0143385701001079
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. MR 1146241
Wen Huang and Xiangdong Ye, Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos, Topology Appl. 117 (2002), no. 3, 259–272. MR 1874089, DOI 10.1016/S0166-8641(01)00025-6
K. Janková and J. Smítal, A characterization of chaos, Bull. Austral. Math. Soc. 34 (1986), no. 2, 283–292. MR 854575, DOI 10.1017/S0004972700010157
I. Kan, A chaotic function possessing a scrambled set with positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), no. 1, 45–49. MR 749887, DOI 10.1090/S0002-9939-1984-0749887-4
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. MR 2136308, DOI 10.1007/s11253-005-0055-4
Shi Hai Li, $\omega$-chaos and topological entropy, Trans. Amer. Math. Soc. 339 (1993), no. 1, 243–249. MR 1108612, DOI 10.1090/S0002-9947-1993-1108612-8
T. Y. Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985–992. MR 385028, DOI 10.2307/2318254
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. MR 2021982
Jaume Llibre and MichałMisiurewicz, Horseshoes, entropy and periods for graph maps, Topology 32 (1993), no. 3, 649–664. MR 1231969, DOI 10.1016/0040-9383(93)90014-M
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. MR 889254, DOI 10.1007/978-3-642-70335-5
Mario Martelli, Mai Dang, and Tanya Seph, Defining chaos, Math. Mag. 71 (1998), no. 2, 112–122. MR 1706086, DOI 10.2307/2691012
MichałMisiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 2, 167–169 (English, with Russian summary). MR 542778
MichałMisiurewicz, Chaos almost everywhere, Iteration theory and its functional equations (Lochau, 1984) Lecture Notes in Math., vol. 1163, Springer, Berlin, 1985, pp. 125–130. MR 829765, DOI 10.1007/BFb0076425
Marston Morse and Gustav A. Hedlund, Symbolic Dynamics, Amer. J. Math. 60 (1938), no. 4, 815–866. MR 1507944, DOI 10.2307/2371264
Marston Morse and Gustav A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62 (1940), 1–42. MR 745, DOI 10.2307/2371431
Elena Murinová, Generic chaos in metric spaces, Acta Univ. M. Belii Ser. Math. 8 (2000), 43–50. MR 1857201
Piotr Oprocha, Relations between distributional and Devaney chaos, Chaos 16 (2006), no. 3, 033112, 5. MR 2265261, DOI 10.1063/1.2225513
Piotr Oprocha, Specification properties and dense distributional chaos, Discrete Contin. Dyn. Syst. 17 (2007), no. 4, 821–833. MR 2276477, DOI 10.3934/dcds.2007.17.821
Piotr Oprocha and PawełWilczyński, Shift spaces and distributional chaos, Chaos Solitons Fractals 31 (2007), no. 2, 347–355. MR 2259760, DOI 10.1016/j.chaos.2005.09.069
RafałPikuła, On some notions of chaos in dimension zero, Colloq. Math. 107 (2007), no. 2, 167–177. MR 2284159, DOI 10.4064/cm107-2-1
Józef Piórek, On the generic chaos in dynamical systems, Univ. Iagel. Acta Math. 25 (1985), 293–298. MR 837847
David Ruelle and Floris Takens, On the nature of turbulence, Comm. Math. Phys. 20 (1971), 167–192. MR 284067, DOI 10.1007/BF01646553
Sylvie Ruette, Dense chaos for continuous interval maps, Nonlinearity 18 (2005), no. 4, 1691–1698. MR 2150350, DOI 10.1088/0951-7715/18/4/015
B. Schweizer, A. Sklar, and J. Smítal, Distributional (and other) chaos and its measurement, Real Anal. Exchange 26 (2000/01), no. 2, 495–524. MR 1844132, DOI 10.2307/44154056
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. MR 1227094, DOI 10.1090/S0002-9947-1994-1227094-X
A. Sklar and J. Smítal, Distributional chaos on compact metric spaces via specification properties, J. Math. Anal. Appl. 241 (2000), no. 2, 181–188. MR 1739200, DOI 10.1006/jmaa.1999.6633
J. Smítal, A chaotic function with some extremal properties, Proc. Amer. Math. Soc. 87 (1983), no. 1, 54–56. MR 677230, DOI 10.1090/S0002-9939-1983-0677230-7
I. Kan, A chaotic function possessing a scrambled set with positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), no. 1, 45–49. MR 749887, DOI 10.1090/S0002-9939-1984-0749887-4
J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), no. 1, 269–282. MR 849479, DOI 10.1090/S0002-9947-1986-0849479-9
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. MR 2182670
Jaroslav Smítal and Marta Štefánková, Distributional chaos for triangular maps, Chaos Solitons Fractals 21 (2004), no. 5, 1125–1128. MR 2047330, DOI 10.1016/j.chaos.2003.12.105
Ľubomír Snoha, Generic chaos, Comment. Math. Univ. Carolin. 31 (1990), no. 4, 793–810. MR 1091377
Ľubomír Snoha, Dense chaos, Comment. Math. Univ. Carolin. 33 (1992), no. 4, 747–752. MR 1240197
Stephen Wiggins, Chaotic transport in dynamical systems, Interdisciplinary Applied Mathematics, vol. 2, Springer-Verlag, New York, 1992. MR 1139113, DOI 10.1007/978-1-4757-3896-4