DCDS
$C^1$-stable shadowing diffeomorphisms
Keonhee Lee Kazumine Moriyasu Kazuhiro Sakai
Let $f$ be a diffeomorphism of a closed $C^\infty$ manifold. In this paper, we define the notion of the $C^1$-stable shadowing property for a closed $f$-invariant set, and prove that $(i)$ the chain recurrent set $R(f)$ of $f$ has the $C^1$-stable shadowing property if and only if $f$ satisfies both Axiom A and the no-cycle condition, and $(ii)$ for the chain component $C_f(p)$ of $f$ containing a hyperbolic periodic point $p$, $C_f(p)$ has the $C^1$-stable shadowing property if and only if $C_f(p)$ is the hyperbolic homoclinic class of $p$.
keywords: hyperbolic set $C^1$-stable shadowing property chain component Axiom A. chain recurrent set shadowing pseudo-orbit homoclinic class
DCDS
Regular maps with the specification property
Kazumine Moriyasu Kazuhiro Sakai Kenichiro Yamamoto
Let $f$ be a $C^1$-regular map of a closed $C^{\infty}$ manifold $M$ and $\Lambda$ be a locally maximal closed invariant set of $f$. We show that $f|_{\Lambda}$ satisfies the $C^1$-stable specification property if and only if $\Lambda$ is a hyperbolic elementary set. We also prove that there exists a residual subset $\mathcal{R}$ in the space of $C^1$-regular maps endowed with the $C^1$-topology such that for $f \in \mathcal{R}$, $f|_{\Lambda}$ satisfies the specification property if and only if $\Lambda$ is a hyperbolic elementary set.
keywords: $C^1$-topology. Specification property regular maps Axiom A elementary sets $C^1$-stable specification property

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