• Nonlinear Functional Analysis and Applications
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• 제28권4호
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• pp.929-942
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• 2023

In this paper, we discuss topological ergodic shadowing property and topological pseudo-orbital specification property of iterated function systems(IFS) on uniform spaces. We show that an 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 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.

• 충청수학회지
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• 제37권2호
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• pp.75-80
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• 2024

In this paper, we study some dynamics of the uniform limits of sequences in dynamical systems on a noncompact metric space. We show that if a sequence of homeomorphisms on a noncompact metric space has the uniform ergodic shadowing property, then the uniform limit also has the ergodic shadowing property. Then we apply this result to nonwandering maps.

• 대한수학회보
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• 제61권4호
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• pp.1137-1159
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• 2024

In the paper, we prove that if a continuous map of a compact uniform space is equicontinuous and pointwise topologically stable, then it is persistent. We also show that if a sequence of uniformly expansive continuous maps of a compact uniform space has a uniform limit and the uniform shadowing property, then the limit is topologically stable. In addition, we introduce the concepts of shadowable points and topologically stable points for a continuous map of a compact topological space and obtain that every shadowable point of an expansive continuous map of a compact topological space is topologically stable.

• 대한수학회지
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• 제60권5호
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• pp.1043-1055
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• 2023

In this paper we study a pointwise version of Walters topological stability in the class of homeomorphisms on a compact metric space. We also show that if a sequence of homeomorphisms on a compact metric space is uniformly expansive with the uniform shadowing property, then the limit is expansive with the shadowing property and so topologically stable. Furthermore, we give examples to illustrate our results.

• 대한수학회보
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• 제59권2호
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• pp.319-343
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• 2022

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.