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Expansiveness has been used to study dynamic systems and has been developed for various forms of expansiveness. In this paper, we introduce the concept of kinematic N-expansiveness for flows on a C∞ compact connected manifold M, which is an extension of N-expansive homeomorphisms. We prove that if a vector field X on M is C1 robustly kinematic N-expansive, then it is quasi-Anosov. Furthermore, we consider the divergence-free vector fields and Hamiltonian systems with the kinematic N-expansive property; then, we study their robustness. [ABSTRACT FROM AUTHOR]

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 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. [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]

Let be a -dimensional (), compact smooth Riemannian manifold endowed with a symplectic form . In this paper, we show that, if a symplectic diffeomorphism is -robustly measure expansive, then it is Anosov and a generic measure expansive symplectic diffeomorphism is mixing Anosov. Moreover, for a Hamiltonian systems, if a Hamiltonian system is robustly measure expansive, then is Anosov. [ABSTRACT FROM AUTHOR]

Let f be a diffeomorphism on a C∞ closed surface. In this paper, we show that if f has the C2-stably limit shadowing property, then we have the following: (i) f satisfies the Kupka-Smale condition; (ii) if P(f) is dense in the nonwandering set Ω(f) and if there is a dominated splitting on Ps(f), then f satisfies both Axiom A and the strong transversality condition. [ABSTRACT FROM AUTHOR]