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Let M be a closed n-dimensional smooth Riemannian manifold, and let X be a $C^1$ -vector field of $M.$ Let $\gamma $ be a hyperbolic closed orbit of $X.$ In this paper, we show that X has the $C^1$ -stably shadowing property on the chain component $C_X(\gamma)$ if and only if $C_X(\gamma)$ is the hyperbolic homoclinic class. [ABSTRACT FROM AUTHOR]

Shadowing property and structural stability are important dynamics with close relationship. Pilyugin and Tikhomirov proved that Lipschitz shadowing property implies the structural stability[5]. Todorov gave a similar result that Lipschitz two-sided limit shadowing property also implies structural stability for diffeomorpshisms[10]. In this paper, we define a generalized Lipschitz shadowing property which unifies these two kinds of Lipschitz shadowing properties, and prove that if a diffeomorphism of a compact smooth manifold has this generalized Lipschitz shadowing property then it is structurally stable. The only if part is also considered. [ABSTRACT FROM AUTHOR]

Let X be a C1 vector field of a compact connected boundaryless smooth smooth Riemannian manifold M with dimM ≥ 3. We show the following: (i) if the chain recurrence set (X) is robustly measure expansive, then it is Axiom A without cycles; and (ii) if the homoclinic class H(Orb(p), X) that contains a hyperbolic periodic orbit Orb(p) is generic-robustly measure expansive, then it is hyperbolic. [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 be a symplectic diffeomorphism on a closed -dimensional Riemannian manifold . In this paper, we show that is Anosov if any of the following statements holds: belongs to the -interior of the set of symplectic diffeomorphisms satisfying the limit shadowing property or belongs to the -interior of the set of symplectic diffeomorphisms satisfying the limit weak shadowing property or belongs to the -interior of the set of symplectic diffeomorphisms satisfying the s-limit shadowing property. [ABSTRACT FROM AUTHOR]