Shadowing, expansiveness and specification for C1-conservative systems

Abstract

Abstract. We prove that a C 1 -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, as in [9,21], we prove that the C 1 -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.

Orbital Shadowing for -Generic Volume-Preserving Diffeomorphisms

“…K. Lee and M. Lee [6] proved that a conservative diffeomorphism is in the 1interior of the set of all conservative diffeomorphisms having the orbital shadowing property if and only if it is Anosov. Our result is a generalization of the result in [7]. Let be a closed ∞ Riemannian manifold endowed with a volume form .…”

“…Denote by ( ) the set of all periodic points of . The following was proved by [7]. Since the paper is still not published yet, we give the proof for completeness.…”

Vector fields satisfying the barycenter property

“…Bessa et al [19] proved that C generically, if a divergence free vector eld X has the shadowing property(expansive, speci cation property) then it is Anosov. From the results, we are going to prove C generic divergence free vector elds when it has the barycenter property.…”

Continuum-wise expansive symplectic diffeomorphisms

“…In [5] Bessa, Lee and Wen showed that a volumepreserving diffeomorphism f is C 1 -robustly expansive if and only if f is Anosov. Bessa et al [8] shoed that if a Hamiltonian system is C 2 -robustly expansive then it is Anosov.…”