Abstract
In this paper, we define and study strong Kato chaos for a group action on a compact metric space. Let X be a compact metric space without isolated points, and let G be a topologically commutative group on X. If the dynamical system (X, G) is weakly mixing, then it is chaotic in the strong sense of Kato.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Cairns, A. Kolganova, and A. Nielsen, Topological transitivity and mixing notions for group actions, Rocky Mountain J. Math. 37 (2007), 371–397.
H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Syst. Theory 1 (1967), 1–49.
R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Second edition, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.
H. Kato, Everywhere chaotic homeomorphisms on manifolds and \(k\)-dimensional menger mainfolds, Topology Appl. 72 (1996), 1–17.
E. Kontorovich and M. Megrelishvili, A note on sensitivity of semigroup actions, Semigroup Forum 76 (2008), 33–141.
T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985–992.
M. Martelli, Introduction to Discrete Dynamical Systems and Chaos, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 1999.
F. Polo, Sensitive dependence on initial conditions and chaotic group actions, Proc. Amer. Math. Soc. 138, (2010), 2815–2826.
D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys. 20 (1971), 167–192.
O. V. Rybak, Li–Yorke sensitivity for semigroup actions, Ukrainian Math. J. 65 (2013), 752–759.
L. Wang, G. Huang, and S. Huan, Distributional chaos in a sequence, Nonlinear Anal. 67 (2007), 2131–2136.
L. Wang, J. Liang, Y. Wang, and X. Sun, Transitivity, weakly mixing property and chaos, Modern Physics Letters B. 30 (2016), 1550274 (8 pages).
X. Wu, A remark on topological sequence entropy, Int. J. Bifurcation and Chaos (accepted for publication).
X. Wu, Chaos of transformations induced onto the space of probability measures, Int. J. Bifurcation and Chaos 26 (2016), 1650227 (12 pages).
X. Wu and G. Chen, On the large deviations theorem and ergodicity, Commun. Nonlinear Sci. Numer. Simul. 30 (2016), 243–247.
X. Wu and G. Chen, Scrambled sets of shift operators, J. Nonlinear Sci. Appl. 9 (2016), 2631–2637.
X. Wu and G. Chen, Sensitivity and transitivity of fuzzified dynamical systems, Inform. Sci. 396 (2017), 14–23.
X. Wu, P. Oprocha, and G. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity 29 (2016), 1942–1972.
X. Wu, J. Wang, and G. Chen, \({\fancyscript {F}}\)-sensitivity and multi-sensitivity of hyperspatial dynamical systems, J. Math. Anal. Appl. 429 (2015), 16–26.
X. Wu and X. Wang, On the iteration properties of large deviations theorem, Int. J. Bifurcation and Chaos 26 (2016), 1650054 (6 pages).
X. Wu, L. Wang, and J. Liang, The chain properties and average shadowing property of iterated function systems, Qual. Theory Dyn. Syst. doi:10.1007/s12346-016-0220-1.
X. Wu, L. Wang, and G. Chen, Weighted backward shift operators with invariant distributionally scrambled subsets, Ann. Funct. Anal. 8 (2017), 199–210.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, L., Liang, J. & Chu, Z. Weakly mixing property and chaos. Arch. Math. 109, 83–89 (2017). https://doi.org/10.1007/s00013-017-1044-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-017-1044-1