Mário Bessa

Generic Dynamics of 4-Dimensional C2 Hamiltonian Systems

We study the dynamical behaviour of Hamiltonian flows defined on 4-dimensional compact symplectic manifolds. We find the existence of a C2-residual set of Hamiltonians for which there is an open mod 0 dense set of regular energy surfaces each either being Anosov or having zero Lyapunov exponents almost everywhere. This is in the spirit of the Bochi-Mañé dichotomy for area-preserving diffeomorphisms on compact surfaces [2] and its continuous-time version for 3-dimensional volume-preserving flows [1].

SHADOWING, EXPANSIVENESS AND SPECIFICATION FOR C1-CONSERVATIVE SYSTEMS

We prove that a C1-generic volume-preserving dynamical system (diffeomorphism or flow) has the shadowing property or is expansive or has the weak specification property if and only if it is Anosov. Finally, we prove that the C1-robustness, within the volume-preserving context, of the expansiveness property and the weak specification property, imply that the dynamical system (diffeomorphism or flow) is Anosov.

On the stability of the set of hyperbolic closed orbits of a Hamiltonian

Let H be a Hamiltonian, e ? H(M) ? R and ?H, e a connected component of H-1({e}) without singularities. A Hamiltonian system, say a triple (H, e, ?H, e), is Anosov if ?H, e is uniformly hyperbolic. The Hamiltonian system (H, e, ?H, e) is a Hamiltonian star system if all the closed orbits of ?H, e are hyperbolic and the same holds for a connected component of -1({?}), close to ?H, e, for any Hamiltonian , in some C2-neighbourhood of H, and ? in some neighbourhood of e.

Shades of hyperbolicity for Hamiltonians

We prove that a Hamiltonian system is globally hyperbolic if any of the following statements hold: H is robustly topologically stable; H is stably shadowable; H is stably expansive; and H has the stable weak specification property. Moreover, we prove that, for a C2-generic Hamiltonian H, the union of the partially hyperbolic regular energy hypersurfaces and the closed elliptic orbits, forms a dense subset of M. As a consequence, any robustly transitive regular energy hypersurface of a C2-Hamiltonian is partially hyperbolic. Finally, we prove that stable weakly-shadowable regular energy hypersurfaces are partially hyperbolic.

Abundance of elliptic dynamics on conservative three-flows

We consider a compact three-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C 1-residual (dense G δ) such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M. This is the flow-setting counterpart of Newhouses Theorem 1.3 (S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Am. J. Math. 99 (1977), pp. 1061–1087). Our result follows from two theorems, the first one says that if Λ is a hyperbolic invariant set for some class C 1 zero divergence vector field X on M, then either X is Anosov, or else Λ has empty interior. The second one says that, if X is not Anosov, then for any open set U ⊆ M there exists Y arbitrarily close to X such that Y t has an elliptical closed orbit through U.

HAMILTONIAN ELLIPTIC DYNAMICS ON SYMPLECTIC 4-MANIFOLDS

We consider C 2 -Hamiltonian functions on compact 4-dimensional symplectic manifolds to study the 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 C 2 -generic Hamiltonian, the elliptic closed orbits are generic.

Dynamics of Generic Multidimensional Linear Differential Systems

Abstract We prove that there exists a residual subset R (with respect to the C0 topology) of d-dimensional linear differential systems based in a μ-invariant flow and with transition matrix evolving in GL(d,ℝ) such that if A ∈ R, then, for μ-a.e. point, the Oseledets splitting along the orbit is dominated (uniform projective hyperbolicity) or else the Lyapunov spectrum is trivial. Moreover, in the conservative setting, we obtain the dichotomy: dominated splitting versus zero Lyapunov exponents.

Dominated splitting and zero volume for incompressible three flows

We prove that there exists an open and dense subset of the incompressible 3-flows of class C2 such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincare flow, then it must be an Anosov flow. With this result we are able to extend the dichotomies of Bochi–Mane (see Bessa 2007 Ergod. Theory Dyn. Syst. 27 1445–72, Bochi 2002 Ergod. Theory Dyn. Syst. 22 1667–96, Mane 1996 Int. Conf. on Dynamical Systems (Montevideo, Uruguay, 1995) (Harlow: Longman) pp 110–9) and of Newhouse (see Newhouse 1977 Am. J. Math. 99 1061–87, Bessa and Duarte 2007 Dyn. Syst. Int. J. submitted Preprint 0709.0700) for flows with singularities. That is, we obtain for a residual subset of the C1 incompressible flows on 3-manifolds that: (i) either all Lyapunov exponents are zero or the flow is Anosov and (ii) either the flow is Anosov or else the elliptic periodic points are dense in the manifold.

A remark on the topological stability of symplectomorphisms

Abstract We prove that the C 1 interior of the set of all topologically stable C 1 symplectomorphisms is contained in the set of Anosov symplectomorphisms.

Removing zero Lyapunov exponents in volume-preserving flows

Baraviera and Bonatti (2003 Ergod. Theory Dyn. Syst. 23 1655–70) proved that it is possible to perturb, in the C1-topology, a stably ergodic, volume-preserving and partially hyperbolic diffeomorphism in order to obtain a non-zero sum of all the Lyapunov exponents in the central direction. In this paper we obtain the analogous result for volume-preserving flows.