CHAIN COMPONENTS WITH THE STABLE SHADOWING PROPERTY FOR C1 VECTOR FIELDS

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.

Asymptotic measure-expansiveness for generic diffeomorphisms

In this paper, we will assume M M to be a compact smooth manifold and f : M → M f:M\to M to be a diffeomorphism. We herein demonstrate that a C 1 {C}^{1} generic diffeomorphism f f is Axiom A and has no cycles if f f is asymptotic measure expansive. Additionally, for a C 1 {C}^{1} generic diffeomorphism f f , if a homoclinic class H ( p , f ) H\left(\hspace{0.08em}p,f) that contains a hyperbolic periodic point p p of f f is asymptotic measure-expansive, then H ( p , f ) H\left(\hspace{0.08em}p,f) is hyperbolic of f f .

Measure-Expansive Homoclinic Classes for C1 Generic Flows

In this paper, we prove that for a generically C1 vector field X of a compact smooth manifold M, if a homoclinic class H(γ,X) which contains a hyperbolic closed orbit γ is measure expansive for X then H(γ,X) is hyperbolic.

There are no proper topological hyperbolic homoclinic classes for area-preserving maps

We begin by defining a homoclinic class for homeomorphisms. Then we prove that if a topological homoclinic class Λ associated with an area-preserving homeomorphism f on a surface M is topologically hyperbolic (i.e. has the shadowing and expansiveness properties), then Λ = M and f is an Anosov homeomorphism.

Lyapunov stable homoclinic classes for smooth vector fields

In this paper, we show that for generic C1, if a flow Xt has the shadowing property on a bi-Lyapunov stable homoclinic class, then it does not contain any singularity and it is hyperbolic.

Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes

We prove that, for $C^{1}$-generic diffeomorphisms, if a homoclinic class is not hyperbolic, then there is a non-trivial non-hyperbolic ergodic measure supported on it. This proves a conjecture by Díaz and Gorodetski.

Hyperbolicity versus weak periodic orbits inside homoclinic classes

We prove that, for$C^{1}$-generic diffeomorphisms, if the periodic orbits contained in a homoclinic class$H(p)$have all their Lyapunov exponents bounded away from zero, then$H(p)$must be (uniformly) hyperbolic. This is in the spirit of the works on the stability conjecture, but with a significant difference that the homoclinic class$H(p)$is not known isolated in advance, hence the ‘weak’ periodic orbits created by perturbations near the homoclinic class have to be guaranteed strictly inside the homoclinic class. In this sense the problem is of an ‘intrinsic’ nature, and the classical proof of the stability conjecture does not work. In particular, we construct in the proof several perturbations which are not simple applications of the connecting lemmas.

HYPERBOLICITY OF HOMOCLINIC CLASSES OF VECTOR FIELDS

Let${\it\gamma}$be a hyperbolic closed orbit of a$C^{1}$vector field$X$on a compact$C^{\infty }$manifold$M$and let$H_{X}({\it\gamma})$be the homoclinic class of$X$containing${\it\gamma}$. In this paper, we prove that if a$C^{1}$-persistently expansive homoclinic class$H_{X}({\it\gamma})$has the shadowing property, then$H_{X}({\it\gamma})$is hyperbolic.