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In this paper, we introduce the notion of the asymptotic orbital pseudo-orbit tracing property for a vector field X of a compact smooth manifold M. We show that if a vector field X has the C 1 robust asymptotic orbital pseudo-orbit tracing property, then it is Anosov, and if a C 1 generic vector field X has the asymptotic orbital pseudo-orbit tracing property, then it is Anosov. Moreover, we also show our results for divergence-free vector fields and Hamiltonian systems. [ABSTRACT FROM AUTHOR]

Let γ be a hyperbolic closed orbit of a C vector field X on a compact C manifold M of dimension n ≥ 3, and let H (γ) be the homoclinic class of X containing γ. In this paper, we prove that C -generically, if H (γ) is expansive and isolated, then it is hyperbolic. [ABSTRACT FROM AUTHOR]