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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 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]