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


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.