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Distributional chaos in multifractal analysis, recurrence and transitivity

  • AN CHEN (a1) and XUETING TIAN (a1)

Abstract

There is much research on the dynamical complexity on irregular sets and level sets of ergodic average from the perspective of density in base space, the Hausdorff dimension, Lebesgue positive measure, positive or full topological entropy (and topological pressure), etc. However, this is not the case from the viewpoint of chaos. There are many results on the relationship of positive topological entropy and various chaos. However, positive topological entropy does not imply a strong version of chaos, called DC1. Therefore, it is non-trivial to study DC1 on irregular sets and level sets. In this paper, we will show that, for dynamical systems with specification properties, there exist uncountable DC1-scrambled subsets in irregular sets and level sets. Meanwhile, we prove that several recurrent level sets of points with different recurrent frequency have uncountable DC1-scrambled subsets. The major argument in proving the above results is that there exists uncountable DC1-scrambled subsets in saturated sets.

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[1] Akin, E. and Kolyada, S.. Li–Yorke sensitivity. Nonlinearity 16 (2003), 14211433.
[2] Ashwin, P., Aston, P. J. and Nicol, M.. On the unfolding of a blowout bifurcation. Phys. D 111(1–4) (1998), 8195.
[3] Ashwin, P. and Field, M.. Heteroclinic networks in coupled cell systems. Arch. Ration. Mech. Anal. 148(2) (1999), 107143.
[4] Barreira, L.. Thermodynamic Formalism and Applications to Dimension Theory. Springer Science & Business Media, 2011.
[5] Barreira, L. and Doutor, P.. Almost additive multi-fractal analysis. J. Math. Pures Appl. (9) 92 (2009), 117.
[6] Barreira, L. and Saussol, B.. Variational principles and mixed multifractal spectra. Trans. Amer. Math. Soc. 353(10) (2001), 39193944.
[7] Barreira, L. and Schmeling, J.. Sets of non-typical points have full topological entropy and full Hausdorff dimension. Israel J. Math. 116 (2000), 2970.
[8] Blanchard, F., Glasner, E., Kolyada, S. and Maass, A.. On Li–Yorke pairs. J. Reine Angew. Math. 547 (2002), 5168.
[9] Blanchard, F., Huang, W. and Snoha, L.. Topological size of scrambled sets. Colloq. Math. 110 (2008), 293361.
[10] Bowen, R.. Periodic points and measures for axiom a diffeomorphisms. Trans. Amer. Math. Soc. 154 (1971), 377397.
[11] Bruckner, A. M. and Hu, T.. On scrambled sets for chaotic functions. Trans. Amer. Math. Soc. 301 (1987), 289297.
[12] Buzzi, J.. Specification on the interval. Trans. Amer. Math. Soc. 349(7) (1997), 27372754.
[13] Chen, E., Kupper, T. and Shu, L.. Topological entropy for divergence points. Ergod. Th. & Dynam. Sys. 25(4) (2005), 11731208.
[14] Climenhaga, V.. Topological pressure of simultaneous level sets. Nonlinearity 26(1) (2013), 241268.
[15] Climenhaga, V., Thompson, D. and Yamamoto, K.. Large deviations for systems with non-uniform structure. Trans. Amer. Math. Soc. 369(6) (2017), 41674192.
[16] Dateyama, M.. Invariant measures for homeomorphisms with almost weak specification. Tokyo J. Math. 04 (1981), 9396.
[17] Denker, M., Grillenberger, C. and Sigmund, K.. Ergodic Theory on Compact Spaces (Lecture Notes in Mathematics, 527) . Springer, Berlin, 1976, p. 177.
[18] Devaney, R.. A First Course in Chaotic Dynamical Systems. Perseus Books, 1992.
[19] Dong, Y., Oprocha, P. and Tian, X.. On the irregular points for systems with the shadowing property. Ergod. Th. & Dynam. Sys. 38(6) (2018), 21082131.
[20] Dong, Y. and Tian, X.. Different statistical future of dynamical orbits over expanding or hyperbolic systems (I): empty syndetic center. Preprint, 2017, arXiv:1701.01910v2.
[21] Dong, Y. and Tian, X.. Different statistical future of dynamical orbits over expanding or hyperbolic systems (II): nonempty syndetic center. Preprint, 2018, arXiv:1803.06796.
[22] Downarowicz, T.. Positive topological entropy implies chaos DC2. Proc. Amer. Math. Soc. 142(1) (2012), 137149.
[23] Eizenberg, A., Kifer, Y. and Weiss, B.. Large deviations for Z d -actions. Commun. Math. Phys. 164(3) (1994), 433454.
[24] Fan, A., Feng, D. and Wu, J.. Recurrence, dimensions and entropy. J. Lond. Math. Soc. (2) 64 (2001), 229244.
[25] Feng, D. and Huang, W.. Lyapunov spectrum of asymptotically sub-additive potentials. Commun. Math. Phys. 297(1) (2010), 143.
[26] Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, 1981.
[27] He, W., Yin, J. and Zhou, Z.. On quasi-weakly almost periodic points. Sci. China Math. 56(3) (2013), 597606.
[28] Huang, Y., Tian, X. and Wang, X.. Transitively-saturated property, Banach recurrence and Lyapunov regularity. Nonlinearity 32(7) (2019), 27212757.
[29] Huang, Y. and Wang, X.. Recurrence of transitive points in dynamical systems with the specification property. Acta Math. Sin. (Engl. Ser.) (2018), 18791891.
[30] Jakobson, M. V.. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81(1) (1981), 3988.
[31] Kan, I.. A chaotic function possessing a scrambled set with positive Lebesgue measure. Proc. Amer. Math. Soc. 92 (1984), 4549.
[32] Kiriki, S. and Soma, T.. Takens’ last problem and existence of non-trivial wandering domains. Adv. Math. 306 (2017), 524588.
[33] Kościelniak, P.. On genericity of shadowing and periodic shadowing property. J. Math. Anal. Appl. 310 (2005), 188196.
[34] Kościelniak, P.. On the genericity of chaos. Topol. Appl. 154 (2007), 19511955.
[35] Koscielniak, P., Mazur, M., Oprocha, P. and Pilarczyk, P.. Shadowing is generic-a continuous map case. Discrete Contin. Dyn. Syst. 34(9) (2014), 35913609.
[36] Kwietniak, D., Oprocha, P. and Rams, M.. On entropy of dynamical systems with almost specification. Israel J. Math. 213(1) (2016), 475503.
[37] Li, T. Y. and Yorke, J. A.. Period three implies chaos. Amer. Math. Monthly 82(10) (1975), 985992.
[38] Liang, C., Liao, G., Sun, W. and Tian, X.. Variational equalities of entropy in nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 369(5) (2017), 31273156.
[39] Liang, C., Sun, W. and Tian, X.. Ergodic properties of invariant measures for C 1+𝛼 non-uniformly hyperbolic systems. Ergod. Th. & Dynam. Sys. 33(2) (2013), 560584.
[40] Nemytskii, V. and Stepanov, V.. Qualitative Theory of Differential Equations, Vol. 2083. Princeton University Press, 2015 (Originally 1960).
[41] Oprocha, P.. Specification properties and dense distributional chaos. Discrete Contin. Dyn. Syst. 17(4) (2007), 821833.
[42] Oprocha, P.. Distributional chaos revisited. Trans. Amer. Math. Soc. 361 (2009), 49014925.
[43] Oprocha, P. and S̆tefánková, M.. Specification property and distributional chaos almost everywhere. Proc. Amer. Math. Soc. 136(11) (2008), 39313940.
[44] Oxtoby, J. C.. Ergodic sets. Bull. Amer. Math. Soc. (N.S.) 58 (1952), 116136.
[45] Parry, W.. On the 𝛽-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11(3–4) (1960), 401416.
[46] Pesin, Y. B.. Dimension Theory in Dynamical Systems: Contemporary Views and Applications. University of Chicago Press, Chicago, IL, 1997.
[47] Pesin, Y. B. and Pitskel’, B.. Topological pressure and the variational principle for noncompact sets. Funct. Anal. Appl. 18 (1984), 307318.
[48] Pfister, C. E. and Sullivan, W. G.. Large deviations estimates for dynamical systems without the specification property. Application to the 𝛽-shifts. Nonlinearity 18(1) (2005), 237261.
[49] Pfister, C. E. and Sullivan, W. G.. On the topological entropy of saturated sets. Ergod. Th. & Dynam. Sys. 27(3) (2007), 929956.
[50] Pikula, R.. On some notions of chaos in dimension zero. Colloq. Math. 107 (2007), 167177.
[51] Rényi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8(3–4) (1979), 477493.
[52] Ruelle, D.. Historic behaviour in smooth dynamical systems. Global Analysis of Dynamical Systems. CRC Press, Boca Raton, FL, 2001, pp. 6366.
[53] Schweizer, B. and Smítal, J.. Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Amer. Math. Soc. 344(2) (1994), 737754.
[54] Sigmund, K.. On dynamical systems with the specification property. Trans. Amer. Math. Soc. 190 (1974), 285299.
[55] Sklar, A. and Smítal, J.. Distributional chaos on compact metric spaces via specification properties. J. Math. Anal. Appl. 241(2) (2000), 181188.
[56] Smítal, J.. Symbolic dynamics for 𝛽-shifts and self-normal numbers. Ergod. Th. & Dynam. Sys. 17(3) (2000), 675694.
[57] Smítal, J. and S̆tefánková, M.. Distributional chaos for triangular maps. Chaos Solitons Fractals 21(5) 11251128.
[58] Takens, F.. Orbits with historic behaviour, or non-existence of averages. Nonlinearity 21 (2008), 3336.
[59] Takens, F. and Verbitski, E.. On the variational principle for the topological entropy of certain non-compact sets. Ergod. Th. & Dynam. Sys. 23(1) (2003), 317348.
[60] Thompson, D.. The irregular set for maps with the specification property has full topological pressure. Dyn. Syst. 25(1) (2008), 2551.
[61] Thompson, D.. A variational principle for topological pressure for certain non-compact sets. J. Lond. Math. Soc. (2) 80(3) (2009), 585602.
[62] Thompson, D.. Irregular sets, the 𝛽-transformation and the almost specification property. Trans. Amer. Math. Soc. 364(10) (2012), 53955414.
[63] Tian, X.. Different asymptotic behaviour versus same dynamicl complexity: recurrence & (ir)regularity. Adv. Math. 288 (2016), 464526.
[64] Tian, X. and Varandas, P.. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete Contin. Dyn. Syst. A 37(10) (2017), 54075431.
[65] Yamamoto, K.. On the weaker forms of the specification property and their applications. Proc. Amer. Math. Soc. 137(11) (2009), 38073814.
[66] Yan, Q., Yin, J. and Wang, T.. A note on quasi-weakly almost periodic point. Acta Math. Sin. (Engl. Ser.) 31(4) (2015), 637646.
[67] Zhou, Z. and Feng, L.. Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: a brief survey of recent results. Nonlinearity 17(2) (2004), 493502.
[68] Zhou, Z. and He, W.. Level of the orbit’s topological structure and topological semi-conjugacy. Sci. China Ser. A 38(8) (1995), 897907.

Keywords

MSC classification

Distributional chaos in multifractal analysis, recurrence and transitivity

  • AN CHEN (a1) and XUETING TIAN (a1)

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