We prove that for a C1-generic (dense Gδ) subset of all the conservative vector fields on three-dimensional compact manifolds without singularities, we have for Lebesgue almost every (a.e.) point p∈M that either the Lyapunov exponents at p are zero or X is an Anosov vector field. Then we prove that for a C1-dense subset of all the conservative vector fields on three-dimensional compact manifolds, we have for Lebesgue a.e. p∈M that either the Lyapunov exponents at p are zero or p belongs to a compact invariant set with dominated splitting for the linear Poincaré flow.

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We consider a compact 3-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 1residual (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 flowsetting counterpart of Newhouse's Theorem 1.3 [17]. Our result follows from two theorems, the first one is the 3-dimensional continuous-time version of a theorem of Xia [21] and 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 is a version, for our 3-dimensional class, of Theorem 2 of Saghin-Xia [20] and 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 .

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We study the dynamical behaviour of Hamiltonian flows defined on 4-dimensional compact symplectic manifolds. We find the existence of a C 2 -residual set of Hamiltonians for which there is an open mod 0 dense set of regular energy surfaces each being either 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].

We analyze the Lyapunov spectrum of the relative Ruelle operator associated with a skew product whose base is an ergodic automorphism and whose fibers are full shifts. We prove that these operators can be approximated in the $C^0$-topology by positive matrices with an associated dominated splitting.Comment: The article now contains a section on decay of correlations of relative transfer operator

Link to this article: http://journals.cambridge.org/abstract_S0305004110000253 How to cite this article: MÁRIO BESSA, CÉLIA FERREIRA and JORGE ROCHA (2010). On the stability of the set of hyperbolic closed orbits of a Hamiltonian. Abstract Let H be a Hamiltonian, e ∈ H (M) ⊂ R and E H,e a connected component of H −1 ({e}) without singularities. A Hamiltonian system, say a triple (H, e, E H,e ), is Anosov if E H,e is uniformly hyperbolic. The Hamiltonian system (H, e, E H,e ) is a Hamiltonian star system if all the closed orbits of E H,e are hyperbolic and the same holds for a connected component of H −1 ({ẽ}), close to E H,e , for any HamiltonianH , in some C 2 -neighbourhood of H , andẽ in some neighbourhood of e.In this paper we show that a Hamiltonian star system, defined on a four-dimensional symplectic manifold, is Anosov. We also prove the stability conjecture for Hamiltonian systems on a four-dimensional symplectic manifold. Moreover, we prove the openness and the structural stability of Anosov Hamiltonian systems defined on a 2d-dimensional manifold, d 2.

Let M be a surface and R : M → M an involution whose set of fixed points is a submanifold with dimension 1 and such that DR x ∈ SL(2, R) for all x. We will show that there is a residual subset of C 1 areapreserving R-reversible diffeomorphisms which are either Anosov or have zero Lyapunov exponents at almost every point.