We define and study expansiveness, shadowing, and topological stability for a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a metric space. 1. Introduction In the recent past, lots of studies have been done regarding dynamical properties in nonautonomous discrete dynamical systems. In [1], Kolyada and Snoha gave definition of topological entropy in nonautonomous discrete systems. In [2], Kolyada et al. discussed minimality of nonautonomous dynamical systems. In [3, 4], authors studied -limit sets in nonautonomous discrete systems, respectively. In [5], Krabs discussed stability and controllability in nonautonomous discrete systems. In [6, 7], Huang et al. studied topological pressure and preimage entropy of nonautonomous discrete systems. In [8–15], authors studied chaos in nonautonomous discrete systems. In [16], Liu and Chen studied -limit sets and attraction of nonautonomous discrete dynamical systems. In [8, 17] authors studied weak mixing and chaos in nonautonomous discrete systems. In [18] Yokoi studied recurrence properties of a class of nonautonomous discrete systems. Recently in [19] we defined and studied expansiveness, shadowing, and topological stability in nonautonomous discrete dynamical systems given by a sequence of continuous maps on a metric space. In this paper we define and study expansiveness, shadowing, and topological stability in nonautonomous discrete dynamical systems given by a sequence of homeomorphisms on a metric space. In the next section, we define and study expansiveness of a time varying homeomorphism on a metric space. In section following to the next section, we define and study shadowing or P.O.T.P. for a time varying homeomorphism on a metric space. In the final section, we study topological stability of a time varying homeomorphism on a compact metric space. 2. Expansiveness of a Nonautonomous Discrete System Induced by a Sequence of Homeomorphisms Throughout this paper we consider to be a metric space and to be a sequence of homeomorphisms, , where we always consider to be the identity map on and to be a time varying homeomorphism on . We denote For any , we define and, for , we define to be the identity map on . For time varying homeomorphism on , its inverse map is given by . For any , we define a time varying map ( th-iterate of ) on , where Thus , for , and, for , . Also, for , , where each is the identity map on . Definition 1. Let be a metric space and a sequence of maps, . For a point , let then the sequence , denoted by , is said to be the orbit of under time varying
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