Chaos for finitely generated semigroup actions
-
2098
Downloads
-
3809
Views
Authors
Lidong Wang
- School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China.
- Department of Mathematics, Dalian Minzu University, Dalian, 116600, People’s Republic of China.
Yingcui Zhao
- School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China.
Zhenyan Chu
- Department of Mathematics, Dalian Minzu University, Dalian, 116600, People’s Republic of China.
Abstract
In this paper, we define and study Li-Yorke chaos and distributional chaos along a sequence for finitely generated semigroup
actions. Let X be a compact space with metric d and G be a semigroup generated by \(f_1, f_2, ..., f_m\) which are finitely many
continuous mappings from X to itself. Then we show if (X,G) is transitive and there exists a common fixed point for all the
above mappings, then (X,G) is chaotic in the sense of Li-Yorke. And we give a sufficient condition for (X,G) to be uniformly
distributionally chaotic along a sequence and chaotic in the strong sense of Li-Yorke. At the end of this paper, an example on
the one-sided symbolic dynamical system for (X,G) to be chaotic in the strong sense of Li-Yorke and uniformly distributionally
chaotic along a sequence is given.
Share and Cite
ISRP Style
Lidong Wang, Yingcui Zhao, Zhenyan Chu, Chaos for finitely generated semigroup actions, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 7, 3843--3850
AMA Style
Wang Lidong, Zhao Yingcui, Chu Zhenyan, Chaos for finitely generated semigroup actions. J. Nonlinear Sci. Appl. (2017); 10(7):3843--3850
Chicago/Turabian Style
Wang, Lidong, Zhao, Yingcui, Chu, Zhenyan. "Chaos for finitely generated semigroup actions." Journal of Nonlinear Sciences and Applications, 10, no. 7 (2017): 3843--3850
Keywords
- Li-Yorke chaos
- distributional chaos along a sequence
- finitely generated semigroup actions.
MSC
References
-
[1]
G. Cairns, A. Kolganova, A. Nielsen, Topological transitivity and mixing notions for group actions, Rocky Mountain J. Math., 37 (2007), 371–397.
-
[2]
E. Kontorovich, M. Megrelishvili , A note on sensitivity of semigroup actions, Semigroup Forum, 76 (2008), 133–141.
-
[3]
F. Polo, Sensitive dependence on initial conditions and chaotic group actions, Proc. Amer. Math. Soc., 138 (2010), 2815– 2826.
-
[4]
O. V. Rybak, Li-Yorke sensitivity for semigroup actions, Ukrainian Math. J., 65 (2013), 752–759.
-
[5]
L.-D. Wang, Y.-N. Li, Y.-L. Gao, H. Liu, Distributional chaos of time-varying discrete dynamical systems, Ann. Polon. Math., 107 (2013), 49–57.
-
[6]
L.-D. Wang, G.-F. Liao, S.-M. Huan, Distributional chaos in a sequence, Nonlinear Anal., 67 (2007), 2131–2136.
-
[7]
X.-X. Wu, Chaos of transformations induced onto the space of probability measures, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 12 pages.
-
[8]
X.-X. Wu, A remark on topological sequence entropy, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 7 pages.
-
[9]
X.-X. Wu, G.-R. Chen, Sensitivity and transitivity of fuzzified dynamical systems, Inform. Sci., 396 (2017), 14–23.
-
[10]
X.-X. Wu, P. Oprocha, G.-R. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity, 29 (2016), 1942–1972.
-
[11]
X.-X. Wu, X. Wang, On the iteration properties of large deviations theorem, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 6 pages.
-
[12]
X.-X. Wu, J.-J. Wang, G.-R. Chen, F-sensitivity and multi-sensitivity of hyperspatial dynamical systems, J. Math. Anal. Appl., 429 (2015), 16–26.
-
[13]
X.-X. Wu, X. Wang, G.-R. Chen, On the large deviations of weaker types, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 12 pages.
-
[14]
X.-X. Wu, L.-D. Wang, G.-R. Chen, Weighted backward shift operators with invariant distributionally scrambled subsets, Ann. Funct. Anal., 8 (2017), 199–210.
-
[15]
X.-X. Wu, L.-D. Wang, J.-H. Liang, The chain properties and average shadowing property of iterated function systems, Qual. Theory Dyn. Syst., 2016 (2016), 9 pages.
-
[16]
X.-X. Wu, L.-D. Wang, J.-H. Liang, The chain properties and Li-Yorke sensitivity of zadehs extension on the space of upper semi-continuous fuzzy sets, Iran. J. Fuzzy Syst., (Accepted),
-
[17]
J.-C. Xiong, Chaos in a topologically transitive system, Sci. China Ser. A, 48 (2005), 929–939.