Kazuhiro Sakai

Various shadowing properties for positively expansive maps

Abstract In this paper, various shadowing properties are considered for a positively expansive map on a compact metrizable space. We show that the Lipschitz shadowing property, the s -limit shadowing property and the strong shadowing property are all equivalent to the (usual) shadowing property for a positively expansive map. Furthermore, for a positively expansive open map, the average shadowing property is shown.

Diffeomorphisms satisfying the specification property

Let f be a diffeomorphism of a closed C ∞ manifold M. In this paper, we introduce the notion of the C 1 -stable specification property for a closed f-invariant set Λof M, and we prove that f/ Λ satisfies a C 1 -stable specification property if and only if Λ is a hyperbolic elementary set. As a corollary, the C 1 -interior of the set of diffeomorphisms of M satisfying the specification property is characterized as the set of transitive Anosov diffeomorphisms.

Vector fields with topological stability

In this paper, we give a characterization of the structurally stable vector fields by making use of the notion of topological stability. More precisely, it is proved that the C1 interior of the set of all topologically stable C1 vector fields coincides with the set of all vector fields satisfying Axiom A and the strong transversality condition.

C0 transversality and shadowing properties

Let f be an Axiom A diffeomorphism of a closed smooth two-dimensional manifold. It is shown that the following statements are equivalent: (a) f satisfies the C0 transversality condition, (b) f has the shadowing property, and (c) f has the inverse shadowing property with respect to a class of continuous methods.

Shadowing properties of L-hyperbolic homeomorphisms

Abstract We introduce the notion of L -hyperbolic homeomorphisms on compact metric spaces as a strict generalization of Axiom A diffeomorphisms and prove that the notion is equivalent to expansive homeomorphisms having the shadowing property and to Ruelles Smale spaces. Furthermore, for L -hyperbolic homeomorphisms, both the Lipschitz shadowing property and the average shadowing property are shown.

Continuum-wise expansive diffeomorphisms

In this paper, we show that the

DIFFEOMORPHISMS WITH THE SHADOWING PROPERTY

C^1

Hyperbolic metrics of expansive homeomorphisms

interior of the set of all continuum-wise expansive diffeomorphisms of a closed manifold coincides with the

Detecting topological groups which are (locally) homeomorphic to LF-spaces

C^1

Diffeomorphisms with C 2 stable shadowing

interior of the set of all expansive diffeomorphisms. And the