This paper focus on the chaotic properties of minimal subshift of shift operators. It is proved that the minimal subshift of shift operators is uniformly distributional chaotic, distributional chaotic in a sequence, distributional chaotic of type k (
), and
-distribution.
References
[1]
Abdulgader, A., Ismail, M., Zainal, N., et al. (2015) Enhancement of AES Algorithm Based on Chaotic Maps and Shift Operation for Image Encryption. Journal of Theoretical and Applied Information Technology, 71, 2005-2015. https://api.semanticscholar.org/CorpusID:52221648
[2]
Shahna, K.U. and Mohamed, A. (2020) A Novel Image Encryption Scheme Using Both Pixel Level and Bit Level Permutation with Chaotic Map. Applied Soft Computing, 90, Article ID: 106162. https://doi.org/10.1016/j.asoc.2020.106162
[3]
Archer, K. and Elizalde, S. (2014) Cyclic Permutations Realized by Signed Shifts. Journal of Combinatorics, 1, 1-30. https://doi.org/10.4310/JOC.2014.v5.n1.a1
[4]
Neuman, C.P. (1993) Transformations between Delta and Forward Shift Operator Transfer Function Models. IEEE Transactions on Systems, Man, and Cybernetics, 23, 295-296. https://doi.org/10.1109/21.214790
[5]
Grosse-Erdmann, K.G. (1999) Universal Families and Hypercyclic Operators. Bulletin of the American Mathematical Society, 36, 345-381. https://doi.org/10.1090/S0273-0979-99-00788-0
[6]
Queffelec, M. (1987) Substitution Dynamical Systems-Spectral Analysis. In: Morel, J.-M. and Teissier, B., Eds., Lecture Notes in Math, Vol. 1294, Springer-Verlag, Berlin, Heideberg.
[7]
Wu, X.X. and Zhu, P.Y. (2013) Chaos in a Class of Nonconstant Weighted Shift Operators. International Journal of Bifurcation and Chaos, 23, 1350010. https://doi.org/10.1142/S0218127413500107
Wang, J.J. (2018) On the Invariance Maximal Distributional Chaos of Weight Shift Operators on . Acta Mathematica Scientia, 38, 446-453.
[10]
Jiang, Z.Y., Fa, J.H., Zheng, Y.P., et al. (1993) Chaotic Behaviour of a Class of Symbolic Dynamical Systems. IFAC Proceedings Volumes, 26, 233-236. https://doi.org/10.1016/S1474-6670(17)49116-6
[11]
Liao, G.F. and Fan, Q.J. (1998) Minimal Subshifts Which Display Schweizer-Smital Chaos and Have Zero Topological Entropy. Science in China Series A: Mathematics, 41, 33-38. https://doi.org/10.1007/BF02900769
[12]
Fu, X.C., Fu, Y., Duan, J.Q., et al. (2000) Chaotic Properties of Subshifts Generated by a Nonperiodic Recurrent Orbit. International Journal of Bifurcation and Chaos, 10, 1067-1073. https://doi.org/10.1142/S021812740000075X
[13]
Oprocha, P. and Wilczynski, P. (2007) Shift Spaces and Distributional Chaos. Chaos, Solitons and Fractals, 31, 347-355. https://doi.org/10.1016/j.chaos.2005.09.069
[14]
Ban, J.C., Hu, W.G. and Zhang, Z.F. (2023) The Entropy of Multiplicative Subshifts on Trees. Journal of Differential Equations, 352, 373-397. https://doi.org/10.1016/j.jde.2023.01.025
[15]
Grigorchuk, R., Lenz, D., Nagnibeda, T. and Sell, D. (2022) Subshifts with Leading Sequences, Uniformity of Cocycles and Spectra of Schreier Graphs. Advances in Mathematics, 407, 108550. https://doi.org/10.1016/j.aim.2022.108550
[16]
Balibrea, F., Smital, J. and Stefankova, M. (2005) The Three Versions of Distributional Chaos. Chaos, Solitons and Fractals, 23, 1581-1583. https://doi.org/10.1016/S0960-0779(04)00351-0
[17]
Gu, G.S. and Xiong, J.C. (2004) A Note on the Distribution CHAOS. Journal of South China Normal University, 3, 37-41.
[18]
Tang, J. and Zhou, X.Y. (2005) Sequence Distribution Chaos Nonequivalent to SS Chaos. Journal of Nanjing University of Technology, 27, 60-63.