arXiv subjects

Zhihong Xia

Publications and source records attributed to Zhihong Xia.

23 records · Page 2Linked to original sources

arXiv2006

Topological Entropy and Partially Hyperbolic Diffeomorphisms

We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique non-trivial homologies. We prove the following two results: if the center foliation is one dimensional, then the topological entropy is locally a constant; and if the center foliation is two dimensional, then the topological entropy is continuous on the set of all $C^\8$ diffeomorphisms. The proof uses a topological invariant we introduced; Yomdin's theorem on upper semi-continuity; Katok's theorem on lower semi-continuity for two dimensional systems and a refined Pesin-Ruelle inequality we proved for partially hyperbolic diffeomorphisms.

math.DS↗

arXiv2006

Homoclinic Points For Area-Preserving Surface Diffeomorphisms

We show a $C^r$ connecting lemma for area-preserving surface diffeomorphisms and for periodic Hamiltonian on surfaces. We prove that for a generic $C^r$, $r=1, 2, ...$, $\infty$, area-preserving diffeomorphism on a compact orientable surface, homotopic to identity, every hyperbolic periodic point has a transversal homoclinic point. We also show that for a $C^r$, $r=1, 2, ...$, $\infty$ generic time periodic Hamiltonian vector field in a compact orientable surface, every hyperbolic periodic trajectory has a transversal homoclinic point. The proof explores the special properties of diffeomorphisms that are generated by Hamiltonian flows.

math.DS↗

arXiv2005

Hyperbolic Invariant Sets With Positive Measures

If a $C^{1 + a}$, $a >0$, volume-preserving diffeomorphism on a compact manifold has a hyperbolic invariant set with positive volume, then the map is Anosov. We also give a direct proof of ergodicity of volume-preserving $CC^{1+a}$, $a>0$, Anosov diffeomorphism without the usual Hopf argument.

math.DS↗

arXiv2005

Area-Preserving Surface Diffeomorphisms

We prove some generic properties for $C^r$, $r=1, 2, ..., \infty$, area-preserving diffeomorphism on compact surfaces. The main result is that the union of the stable (or unstable) manifolds of hyperbolic periodic points are dense in the surface. This extends the result of Franks and Le Calvez \cite{FL03} on $S^2$ to general surfaces. The proof uses the theory of prime ends and Lefschetz fixed point theorem.

math.DS↗

arXiv2005

A $C^{r}$ Closing Lemma for a Class of Symplectic Diffeomorphisms

We prove a $C^r$ closing lemma for a class of partially hyperbolic symplectic diffeomorphisms. We show that for a generic $C^r$ symplectic diffeomorphism, $r =1, 2, ...,$, with two dimensional center and close to a product map, the set of all periodic points is dense.

math.DS↗