In this paper we discuss the dynamical system induced by sequence of maps i.e. time varying map on a metric space. We define and study shadowing and expansiveness of such dynamical systems. We show that expansiveness and shadowing of time varying maps are conjugacy invariant. Finally, we prove that a time varying map having shadowing and expansiveness is topologically stable in the class of all time varying maps on a compact metric space.Keywords Expansiveness · Shadowing · Conjugacy · Topological stability Mathematics Subject Classification (2010) Primary 54H20; Secondary 37C75 · 37C15
IntroductionExpansiveness and shadowing are very important and useful dynamical properties of maps on metric spaces. They have lots of applications in Topological dynamics, Ergodic theory, Symbolic dynamics and related areas. One can refer [2,20] for detailed study of these notions. The concept of expansiveness originally introduced for homeomorphisms on metric spaces [24]
Beginning with examples, the notion of G-expansiveness over a metric space X on which a topological group G acts is introduced. Some conditions are determined under which expansiveness on X implies G-expansiveness. A characterization of a Gexpansive homeomorphism is obtained which in turn gives a sufficient condition for the homeomorphic extension of a G-expansive homeomorphism to be G-expansive. At the end, some results are stated in the form of concluding remarks.
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.
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