Volume-Preserving Diffeomorphisms with the $$\mathcal {M}_0$$-Shadowing Properties

Abstract

This paper studies the $$\mathcal {M}_0$$ -shadowing property for two types of volume-preserving diffeomorphisms defined on compact manifolds. For symplectic diffeomorphisms defined on symplectic manifolds, the $$C^1$$ -interior of the set of all symplectic diffeomorphisms with the $$\mathcal {M}_0$$ -shadowing property is described by the set of the Anosov diffeomorphisms. If a volume-preserving diffeomorphism in $$\mathrm{Diff}^1_{\mu }(M)$$ is a $$C^1$$ -stable $$\mathcal {M}_0$$ -shadowing diffeomorphism, then M admits a volume-hyperbolic dominated splitting.