• Communications of the Korean Mathematical Society
• /
• v.38 no.1
• /
• pp.243-256
• /
• 2023

We deal with a type of inverse pseudo-orbit tracing property with respect to the class of continuous methods, as suggested and studied by Pilyugin [54]. In this paper, we consider a continuous method induced through the diffeomorphism of a compact smooth manifold, and using the concept, we proved the following: (i) If a diffeomorphism f of a compact smooth manifold M has the robustly pointwise inverse pseudoorbit tracing property, f is structurally stable. (ii) For a C1 generic diffeomorphism f of a compact smooth manifold M, if f has the pointwise inverse pseudo-orbit tracing property, f is structurally stable. (iii) If a diffeomorphism f has the robustly pointwise inverse pseudo-orbit tracing property around a transitive set Λ, then Λ is hyperbolic for f. Finally, (iv) for C¹ generically, if a diffeomorphism f has the pointwise inverse pseudo-orbit tracing property around a locally maximal transitive set Λ, then Λ is hyperbolic for f. In addition, we investigate cases of volume preserving diffeomorphisms.

• Journal of the Chungcheong Mathematical Society
• /
• v.33 no.2
• /
• pp.287-292
• /
• 2020

In this paper, we prove that a distal homeomorphism f of a compact connected metric space X does not have the eventual pseudo orbit tracing property.

• Journal of the Chungcheong Mathematical Society
• /
• v.20 no.3
• /
• pp.343-348
• /
• 2007

In this paper, we show that if a homeomorphism has the pseudo-orbit-tracing-property and its nonwandering set is locally connected, then the limit sets of wandering points whose stable sets have nonempty interior consist of single periodic orbit.

• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.4
• /
• pp.871-878
• /
• 2010

In this paper, we show that if a homeomorphism has the pseudo-orbit-tracing-property and its nonwandering set is locally connected, then the points whose stable sets have nonempty interior are periodic points.

• Journal of the Chungcheong Mathematical Society
• /
• v.35 no.3
• /
• pp.255-267
• /
• 2022

Uniform space is a generalization of metric space. The main purpose of this paper is to extend several results contained in [5, 6] which have for an expansive homeomorphism with the pseudo orbit tracing property(POTP in short) on a compact metric space (X, d) for an expansive homeomorphism with the POTP on a compact uniform space (X, 𝒰). we characterize stable and unstable sets, sink and source and saddle, recurrent points for an expansive homeomorphism which has the POTP on a compact uniform space (X, 𝒰).

• Communications of the Korean Mathematical Society
• /
• v.21 no.2
• /
• pp.355-361
• /
• 2006

We prove that if a continuous surjective map f on a compact metric space X has the average shadowing property, then every point x is chain recurrent. We also show that if a homeomorphism f has more than two fixed points on $S^1$, then f does not satisfy the average shadowing property. Moreover, we construct a homeomorphism on a circle which satisfies the shadowing property but not the average shadowing property. This shows that the converse of the theorem 1.1 in [6] is not true.

• Communications of the Korean Mathematical Society
• /
• v.19 no.4
• /
• pp.759-764
• /
• 2004

Let X = {a} ${\cup}$ {$a_{i}$ ${$\mid$}$i $\in$ N} be a subspace of Euclidean space $E^2$ such that $lim_{{i}{\longrightarrow}{$\infty}}a_{i}$ = a and $a_{i}\;{\neq}\;a_{j}$ for $i{\neq}j$. Then it is well known that the space X has no expansive homeomorphisms with the shadowing property. In this paper we show that the set of all expansive homeomorphisms with the shadowing property on the space Y is dense in the space H(Y) of all homeomorphisms on Y, where Y = {a, b} ${\cup}$ {$a_{i}{$\mid$}i{\in}Z$} is a subspace of $E^2$ such that $lim_{i}$-$\infty$ $a_{i}$ = b and $lim_{{i}{\longrightarrow}{$\infty}}a_{i}$ = a with the following properties; $a_{i}{\neq}a_{j}$ for $i{\neq}j$ and $a{\neq}b$.

• Bulletin of the Korean Mathematical Society
• /
• v.36 no.2
• /
• pp.403-409
• /
• 1999

In this paper we introduce the notions of expansive homeomorphism and its properties, and study the relation between countable compacta and ordinal numbers. Our results extend and improve those of T.Kimura and others.

• Journal of applied mathematics & informatics
• /
• v.7 no.3
• /
• pp.1029-1038
• /
• 2000

In this paper, we show that if x is a positively Lyapunov stable point of an expansive homeomorphism with the pseudo-orbit-tracing-property, then x is a periodic point or its positive limit set consists of only one periodic orbit, and their periods are predictable. We give a necessary and sufficient condition that a basic set is to be a sink or source. Also, we consider some dynamical properties of basic sets.

• Journal of the Korean Mathematical Society
• /
• v.51 no.4
• /
• pp.837-851
• /
• 2014

We consider ${\omega}$-chaos as defined by S. H. Li in 1993. We show that c-dense ${\omega}$-scrambled sets are present in every transitive system with two-sided limit shadowing property (TSLmSP) and that every transitive map on topological graph has a dense Mycielski ${\omega}$-scrambled set. As a preliminary step, we provide a characterization of dynamical properties of maps with TSLmSP.