Manseob Lee

SHADOWING, EXPANSIVENESS AND SPECIFICATION FOR C1-CONSERVATIVE SYSTEMS

We prove that a C1-generic volume-preserving dynamical system (diffeomorphism or flow) has the shadowing property or is expansive or has the weak specification property if and only if it is Anosov. Finally, we prove that the C1-robustness, within the volume-preserving context, of the expansiveness property and the weak specification property, imply that the dynamical system (diffeomorphism or flow) is Anosov.

Stably asymptotic average shadowing property and dominated splitting

Let f be a diffeomorphism of a closed n-dimensional C∞ manifold. In this article, we show that C1-generically, if f has the C1-stably asymptotic average shadowing property on a closed set then it admits a dominated splitting.Mathematical Subject Classification: 34D05; 37C20; 37D30.

Stably inverse shadowable transitive sets and dominated splitting

Let f be a diffeomorphism of a closed n-dimensional smooth manifold. In this paper, we show that if f has the C1-stably inverse shadowing property on a transitive set, then it admits a dominated splitting.

Generic diffeomorphisms with measure-expansive homoclinic classes

We prove that for C1 generic diffeomorphisms, every measure-expansive locally maximal homoclinic class is hyperbolic.

GENERIC DIFFEOMORPHISMS WITH ROBUSTLY TRANSITIVE SETS

Abstract. Let Λ be a robustly transitive set of a diffeomorphism f ona closed C ∞ manifold. In this paper, we characterize hyperbolicity of Λin C 1 -generic sense. 1. IntroductionA fundamental problem in differentiable dynamical systems is to understandhow a robust dynamic property (that is, a property that holds for a system andall C 1 nearby ones) on the underlying manifold would influences the behaviorof the tangent map on the tangent bundle. In this paper, we study the robustdynamic property for a transitive set. Let M be a closed C ∞ manifold, and letDiff(M) be the space of diffeomorphisms of M endowed with the C 1 -topology.Denote by d the distance on M induced from a Riemannian metric k·k on thetangent bundle TM. Let f ∈ Diff(M) and Λ be a closed f-invariant set. Theset Λ is transitive if there is a point x ∈ Λ such that ω(x) = Λ. Here ω(x) isthe forward limit set of x. Denote by f| Λ the restriction of f to the set Λ. Amaximal invariant set of f in an open set U, denoted by Λ f (U), is the set ofpoints whose whole orbit contained in U, that is,Λ

Volume-preserving diffeomorphisms with inverse shadowing

Let f be a volume-preserving diffeomorphism of a closed C∞n-dimensional Riemannian manifold M. In this paper, we prove the equivalence between the following conditions:(a) f belongs to the C1-interior of the set of volume-preserving diffeomorphisms which satisfy the inverse shadowing property with respect to the continuous methods,(b) f belongs to the C1-interior of the set of volume-preserving diffeomorphisms which satisfy the weak inverse shadowing property with respect to the continuous methods,(c) f belongs to the C1-interior of the set of volume-preserving diffeomorphisms which satisfy the orbital inverse shadowing property with respect to the continuous methods,(d) f is Anosov.MSC:37C50, 34D10.

The ergodic shadowing property and homoclinic classes

In this paper, we show that if a diffeomorphism satisfies a local star condition and it has the ergodic shadowing property then it is hyperbolic.MSC:37C29, 37C50.

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.

Chain components with C1-stably orbital shadowing

Let f:M→M be a diffeomorphism on a C∞n-dimensional manifold. Let Cf(p) be the chain component of f associated to a hyperbolic periodic point p. In this paper, we show that (i) if f has the C1-stably orbitally shadowing property on the chain recurrent set R(f), then f satisfies both Axiom A and no-cycle condition, and (ii) if f has the C1-stably orbitally shadowing property on Cf(p), then Cf(p) is hyperbolic.MSC:37C50, 34D10, 37C20, 37C29.Let f : M → M Open image in new window be a diffeomorphism on a C ∞ Open image in new windown-dimensional manifold. Let C f ( p ) Open image in new window be the chain component of f associated to a hyperbolic periodic point p. In this paper, we show that (i) if f has the C 1 Open image in new window-stably orbitally shadowing property on the chain recurrent set R ( f ) Open image in new window, then f satisfies both Axiom A and no-cycle condition, and (ii) if f has the C 1 Open image in new window-stably orbitally shadowing property on C f ( p ) Open image in new window, then C f ( p ) Open image in new window is hyperbolic.

Orbital Shadowing for -Generic Volume-Preserving Diffeomorphisms

We show that -generically, if a volume-preserving diffeomorphism has the orbital shadowing property, then the diffeomorphism is Anosov.