Junmi Park

Chain components with stably limit shadowing property are hyperbolic

Let f be a diffeomorphism on a closed smooth manifold M. In this paper, we show that f has the C1-stably limit shadowing property on the chain component Cf(p) of f containing a hyperbolic periodic point p, if and only if Cf(p) is a hyperbolic basic set.MSC:37C50, 34D10.

Expansive transitive sets for robust and generic diffeomorphisms

ABSTRACT Let f be a diffeomorphism on a compact smooth Riemannian manifold M, and let Λ be a closed f-invariant transitive subset of M. In this paper, we show that f|Λ is C1-stably expansive if and only if Λ is a hyperbolic basic set. Moreover, C1-generically, a locally maximal transitive set Λ is expansive if and only if it is a hyperbolic basic set.

Measure expansive symplectic diffeomorphisms and Hamiltonian systems

Let M be a 2n-dimensional (n ≥ 2), compact smooth Riemannian manifold endowed with a symplectic form ω. In this paper, we show that, if a symplectic diffeomorphism f is C1-robustly measure expansive, then it is Anosov and a C1 generic measure expansive symplectic diffeomorphism f is mixing Anosov. Moreover, for a Hamiltonian systems, if a Hamiltonian system (H,e,ℰH,e) is robustly measure expansive, then (H,e,ℰH,e) is Anosov.

SHADOWABLE CHAIN COMPONENTS AND HYPERBOLICITY

We show that -generically, the shadowable chain component of a -vector field containing a hyperbolic periodic orbit is hyperbolic if it is locally maximal.

Stably Limit Shadowing Diffeomorphisms

Let be a diffeomorphism on a closed surface. In this paper, we show that if has the -stably limit shadowing property, then we have the following: (i) satisfies the Kupka-Smale condition; (ii) if is dense in the nonwandering set () and if there is a dominated splitting on (), then satisfies both Axiom and the strong transversality condition.