Sejal SHAH, Naveenkumar YADAV

Li-Yorke chaos and topological distributional chaos in a sequence

We study here the topological notion of Li-Yorke chaos defined for uniformly continuous self-maps defined on uniform Hausdorff spaces, which are not necessarily compact metrizable. We prove that a weakly mixing uniformly continuous self-map defined on a second countable Baire uniform Hausdorff space without isolated points is Li-Yorke chaotic. Further, we define and study the notion of topological distributional chaos in a sequence for uniformly continuous self-maps defined on uniform Hausdorff spaces. We prove that Li-Yorke chaos is equivalent to topological distributional chaos in a sequence for uniformly continuous self-maps defined on second countable Baire uniform Hausdorff space without isolated points. As a consequence, we obtain that Devaney chaos implies topological distributional chaos in a sequence.

PDF

___

[1] Ahmadi SA, Wu X, Chen G. Topological chain and shadowing properties of dynamical systems on uniform spaces. Topology and its Applications 2020; 275: 107153. doi: 10.1016/j.topol.2020.107153
[2] Arai T. Devaney’s and Li-Yorke’s chaos in uniform spaces. Journal of Dynamical and Control Systems 2018; 24 (1): 93-100. doi: 10.1007/s10883-017-9360-0
[3] Das P, Das T. Various types of shadowing and specification on uniform spaces. Journal of Dynamical and Control Systems 2018; 24 (2): 253-267. doi: 10.1007/s10883-017-9388-1
[4] Devaney RL. An Introduction to Chaotic Dynamical Systems. CA, USA: 2nd edition, Addison-Wesley, 1989.
[5] Gu G, Xiong J. A note on the distribution chaos. Journal of South China Normal University (Natural Science Edition) 2004; No. 3: 37-41. (in Chinese)
[6] Huang W, Ye X. Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos. Topology and its Applications 2002; 117 (3): 259-272. doi: 10.1016/S0166-8641(01)00025-6
[7] Kelley JL. General Topology. NY, USA: D Van Nostrand Company, 1955.
[8] Li J, Oprocha P. On n-scrambled tuples and distributional chaos in a sequence. Journal of Difference Equations and Applications 2013; 19 (6): 927-941. doi: 10.1080/10236198.2012.700307
[9] Li T, Yorke JA. Period three implies chaos. The American Mathematical Monthly 1975; 82 (10): 985-992. doi: 10.1080/00029890.1975.11994008
[10] Oprocha P. Relations between distributional and Devaney chaos. Chaos: An Interdisciplinary Journal of Nonlinear Science 2006; 16 (3): 033112. doi: 10.1063/1.2225513
[11] Schweizer B, Smítal J. Measures of chaos and a spectral decomposition of dynamical systems on the interval. Transactions of the American Mathematical Society 1994; 344 (2): 737-754. doi: 10.1090/S0002-9947-1994-1227094- X
[12] Shah S, Das T, Das R. Distributional chaos on uniform spaces. Qualitative Theory of Dynamical Systems 2020; 19 (1): Paper No. 4. doi: 10.1007/s12346-020-00344-x
[13] Wang L, Huang G, Huan S. Distributional chaos in a sequence. Nonlinear Analysis: Theory, Methods and Applications 2007; 67 (7): 2131-2136. doi: 10.1016/j.na.2006.09.005
[14] Wang L, Yang Y, Chu Z, Liao G. Weakly mixing implies distributional chaos in a sequence. Modern Physics Letters B 2010; 24 (14): 1595-1600. doi: 10.1142/S0217984910023372
[15] Wu X, Ma X, Zhu Z, Lu T. Topological ergodic shadowing and chaos on uniform spaces. International Journal of Bifurcation and Chaos 2018; 28 (3): 1850043. doi: 10.1142/S0218127418500438