On ergodic shadowing and specification properties of nonautonomous discrete dynamical systems

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, University of Sistan and ;Baluchestan, Zahedan, Iran.

10.22103/jmmrc.2021.15567.1114

Abstract

‎We show that a nonautonomous discrete-time dynamical system (NDS) with the ergodic shadowing property is chain mixing‎. ‎As a result‎, ‎it is obtained that a $k$-periodic NDS with the ergodic shadowing property has the shadowing property‎. In particular‎, ‎any $k$-periodic NDS on intervals having the ergodic shadowing is Devaney chaotic‎. Additionally‎, ‎we prove that for an equicontinuous NDS with the shadowing property‎, ‎the notions of topologically mixing‎, ‎pseudo-orbital specification‎, ‎weak specification property‎, ‎and ergodic shadowing property are equivalent‎.

Keywords

References

D.V. Anosov, On a class of invariant sets for smooth dynamical systems, Mathematics Institute Ukrainian Academic Science, vol. 2, Kiev, 1970.
M. L. Blank, Metric properties of -trajectories of dynamical systems with stochastic behaviour, Ergodic Theory Dynam. systems Vol. 8, (1998), 365-378.
R. Bowen, Equilibrium states and ergodic theory of Anosov di eomorphisms, Trans. Amer. Math. Soc., Vol. 154, (1971), 377-397.
E. Camouzis, G. Ladaa, periodically forced Pielou equation, J. Math. Anal. Appl., Vol. 333, (2007), 117-127.
J. Cushing, S. Henson, A periodically forced Beverton-Holt equation, J. Di erence Equ. Appl., Vol. 8, (2002), 1119-1120.
A. Fakhari, F. H. Ghane, On shadowing: ordinary and ergodic,J. Math. Anal. Appl., Vol. 364, (2010), 151-155.
S. Kolyada, L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dyn., Vol. 4, (1996), 205-233.
L. Liu, Y. Sun, Weakly mixing sets and transitive sets for non-autonomous discrete systems, Adv. Di erence Equ., Vol. 2014, (2014), 217-225.
R. Memarbashi, H. Rasuli, Notes on the dynamics of nonautonomous discrete dynamical systems. J. Adv. Res. Dyn. Control Syst., Vol. 6, (2014), 8-17.
H. Parham, F.H. Ghane, E. Rezaali Ergodic shadowing of non-autonomous discrete-time dynamical systems Vol. 9, (2019), 203 -212.
H. Rasuoli, On the shadowing property of nonautonomous discrete systems, Int. J. Nonlinear Anal. Appl., Vol. 7, (2015), 271-277.
M. Salman, R. Das, Speci cation properties for non-autonomous discrete systems, arXiv preprint arXiv:1808.07791, (2018).
M. Salman and R. Das, Dynamics of weakly mixing nonautonomous systems, Internat. J. Bifur. Chaos Appl. Sci. Eng., Vol. 29, (2019), 1950123, 11 pages.
D. Thakkar, R. Das, Topological stability of a sequence of maps on a compact metric space. Bull. Math. Sci., Vol. 4, (2014), 99-111.
C. Tian, G. Chen, Chaos of sequence of maps in a metric space, Chaos Solit. Fract., Vol. 28, (2006), 1067-1075.
P. Walters, on the pseudo orbit tracing property and its relationship to stability, Lecture Notes in Math., Vol. 668, Springer, Berlin, 1978, 224-231.
X. Wu, X. Ma, Z. Zhu, T. Lu, Topological ergodic shadowing and chaos on uniform spaces, Int. J. Bifur. Chaos Appl. Sci. Engrag., Vol. 28, (2018), 1850043, 9 pages.
R.S. Yang, the pseudo orbit tracing property and chaos, Acta Math. Sin., Vol. 39, (1996), 382-386.

History

• Receive Date: 09 March 2020
• Revise Date: 11 January 2021
• Accept Date: 14 January 2021