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Positive expansivity, chain transitivity, rigidity, and specification on general topological spaces
Thiyam Thadoi Devi,Khundrakpam Binod Mangang 대한수학회 2022 대한수학회보 Vol.59 No.2
We discuss the notions of positive expansivity, chain transitivity, uniform rigidity, chain mixing, weak specification, and pseudo orbital specification in terms of finite open covers for Hausdorff topological spaces and entourages for uniform spaces. We show that the two definitions for each notion are equivalent in compact Hausdorff spaces and further they are equivalent to their standard definitions in compact metric spaces. We show that a homeomorphism on a Hausdorff uniform space has uniform $h$-shadowing if and only if it has uniform shadowing and its inverse is uniformly equicontinuous. We also show that a Hausdorff positively expansive system with a Hausdorff shadowing property has Hausdorff $h$-shadowing.
TOPOLOGICAL ERGODIC SHADOWING AND TOPOLOGICAL PSEUDO-ORBITAL SPECIFICATION OF IFS ON UNIFORM SPACES
Thiyam Thadoi Devi,Khundrakpam Binod Mangang,Lalhmangaihzuala 경남대학교 기초과학연구소 2023 Nonlinear Functional Analysis and Applications Vol.28 No.4
In this paper, we discuss topological ergodic shadowing property and topological pseudo-orbital specification property of iterated function systems(\emph{IFS}) on uniform spaces. We show that an \emph{IFS} on a sequentially compact uniform space with topological ergodic shadowing property has topological shadowing property. We define the notion of topological pseudo-orbital specification property and investigate its relation to topological ergodic shadowing property. We find that a topologically mixing \emph{IFS} on a compact and sequentially compact uniform space with topological shadowing property has topological pseudo-orbital specification property and thus has topological ergodic shadowing property.