• 충청수학회지
• /
• 제30권2호
• /
• pp.177-181
• /
• 2017

Let $f:M{\rightarrow}M$ be a diffeomorphism of a compact smooth manifold M. In this paper, we show that $C^1$ generically, if a chain transitive set ${\Lambda}$ is locally maximal then it admits a dominated splitting. Moreover, $C^1$ generically if a chain transitive set ${\Lambda}$ of f is locally maximal then it has zero entropy.

• 충청수학회지
• /
• 제25권4호
• /
• pp.725-730
• /
• 2012

Let $f$ be a diffeomorphism of a closed $C^{\infty}$ manifold M. We show that $C^1$-generically, if $f$ has the shadowing property on a robustly transitive set, then it is hyperbolic.

• 충청수학회지
• /
• 제32권2호
• /
• pp.165-170
• /
• 2019

$C^1$ generically, if a diffeomorphism $f:M{\rightarrow}M$ of a closed smooth Riemannian manifold M has the asymptotic average shadowing property or the average shadowing property then f has a decomposition property.

• 충청수학회지
• /
• 제25권4호
• /
• pp.643-653
• /
• 2012

Let $f\;:\;M\;{\rightarrow}\;M$ be a diffeomorphism on a closed $C^{\infty}$ manifold. Let $\Lambda$ be a transitive set. In this paper, we show that (i) $C^1$-generically, $f$ has the shadowing property on a locally maximal $\Lambda$ if and only if $\Lambda$ is hyperbolic, (ii) f has the $C^1$-stably shadowing property on $\Lambda$ if and only if $\Lambda$ is hyperbolic.

• 대한수학회논문집
• /
• 제24권1호
• /
• pp.127-144
• /
• 2009

Let f be a diffeomorphism of a compact $C^{\infty}$ manifold, and let p be a hyperbolic periodic point of f. In this paper we introduce the notion of $C^1$-stable inverse shadowing for a closed f-invariant set, and prove that (i) the chain recurrent set $\cal{R}(f)$ of f has $C^1$-stable inverse shadowing property if and only if f satisfies both Axiom A and no-cycle condition, (ii) $C^1$-generically, the chain component $C_f(p)$ of f associated to p is hyperbolic if and only if $C_f(p)$ has the $C^1$-stable inverse shadowing property.

• 대한수학회지
• /
• 제55권5호
• /
• pp.1131-1142
• /
• 2018

A notion of measure expansivity for homeomorphisms was introduced by Morales recently as a generalization of expansivity, and he obtained many interesting dynamic results of measure expansive homeomorphisms in [8]. In this paper, we introduce a concept of weak measure expansivity for homeomorphisms which is really weaker than that of measure expansivity, and show that a diffeomorphism f on a compact smooth manifold is $C^1$-stably weak measure expansive if and only if it is ${\Omega}$-stable. Moreover we show that $C^1$-generically, if f is weak measure expansive, then f satisfies both Axiom A and the no cycle condition.

• 대한수학회논문집
• /
• 제27권2호
• /
• pp.369-375
• /
• 2012

In this article, we focus on certain dynamic phenomena in volume-preserving manifolds. Let $M$ be a compact manifold with a volume form ${\omega}$ and $f:M{\rightarrow}M$ be a diffeomorphism of class $\mathcal{C}^1$ that preserves ${\omega}$. In this paper, we do not assume $f$ is $\mathcal{C}^1$-generic. We prove that $f$ satisfies the chain transitivity and we also show that, on $M$, the $\mathcal{C}^1$-stable shadowability is equivalent to the hyperbolicity.