Hamiltonian elliptic dynamics on symplectic 4-manifolds

We consider C2 Hamiltonian functions on compact 4-dimensional symplectic manifolds to study elliptic dynamics of the Hamiltonian flow, namely the so-called Newhouse dichotomy. We show that for any open set U intersecting a far from Anosov regular energy surface, there is a nearby Hamiltonian having an elliptic closed orbit through U. Moreover, this implies that for far from Anosov regular energy surfaces of a C2-generic Hamiltonian the elliptic closed orbits are generic.Comment: 9 page

ATTRACTORS AND ORBIT-FLIP HOMOCLINIC ORBITS FOR STAR FLOWS

Abstract. We study star flows on closed 3-manifolds and prove that they either have a finite number of attractors or can be C 1 approximated by vector fields with orbit-flip homoclinic orbits. 1

BADLY APPROXIMABLE NUMBERS AND VECTORS IN CANTOR-LIKE SETS

Abstract. We show that a large class of Cantor-like sets of R d,d ≥ 1, contains uncountably many badly approximable numbers, respectively badly approximable vectors, when d ≥ 2. An analogous result is also proved for subsets of R d arising in the study of geodesic flows corresponding to (d+1)-dimensional manifolds of constant negative curvature and finite volume, generalizing the set of badly approximable numbers in R. Furthermore, we describe a condition on sets, which is fulfilled by a large class, ensuring a large intersection with these Cantor-like sets. 1

DIFFEOMORPHISMS SATISFYING THE SPECIFICATION PROPERTY

Abstract. Let f be a diffeomorphism of a closed C ∞ manifold M. In this paper, we introduce the notion of the C 1-stable specification property for a closed f-invariant set Λ of M, and we prove that f |Λ satisfies a C 1-stable specification property if and only if Λ is a hyperbolic elementary set. As a corollary, the C 1-interior of the set of diffeomorphisms of M satisfying the specification property is characterized as the set of transitive Anosov diffeomorphisms. 1