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- On C1-robust transitivity of volume-preserving flowsPublication . Bessa, Mário; Rocha, JorgeWe prove that a divergence-free and C1-robustly transitive vector field has no singularities. Moreover, if the vector field is smooth enough then the linear Poincaré flow associated to it admits a dominated splitting over M.
- Homoclinic tangencies versus uniform hyperbolicity for conservative 3-flowsPublication . Bessa, Mário; Rocha, JorgeWe prove that a volume-preserving three-dimensional flow can be C 1 -approximated by a volume-preserving Anosov flow or else by another volume-preserving flow exhibiting a homoclinic tangency. This proves the conjecture of Palis for conservative 3-flows and with respect to the C1-topology.
- Are there chaotic maps in the sphere?Publication . Bessa, MárioTwo of the most popular notions of chaoticity are the one due to Robert Devaney and the one that assumes positive Lyapunov exponents. In this note we discuss the coexistence of both definitions for conservative discrete dynamical systems in the two-sphere and with respect to the C1-generic point of view.
- A generic incompressible flow is topological mixingPublication . Bessa, MárioIn this Note we prove that there exists a residual subset of the set of divergence-free vector fields defined on a compact, connected Riemannian manifold M, such that any vector field in this residual satisfies the following property: Given any two nonempty open subsets U and V of M, there exists τ ∈R such that X^t(U)∩V is non-empty for any t >=τ.
- Dynamics of generic multidimensional linear differential systemsPublication . Bessa, MárioWe prove that there exists a residual subset R (with respect to the C^0 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.
- On the fundamental regions of a fixed point free conservative Hénon mapPublication . Bessa, Mário; Rocha, JorgeIt is well known that an orientation-preserving homeomorphism of the plane without fixed points has trivial dynamics; that is, its non-wandering set is empty and all the orbits diverge to infinity. However, orbits can diverge to infinity in many different ways (or not) giving rise to fundamental regions of divergence. Such a map is topologically equivalent to a plane translation if and only if it has only one fundamental region. We consider the conservative, orientation-preserving and fixed point free Hénon map and prove that it has only one fundamental region of divergence. Actually, we prove that there exists an area-preserving homeomorphism of the plane that conjugates this Hénon map to a translation.
- Dynamics of generic 2-dimensional linear differential systemsPublication . Bessa, MárioWe prove that for a C0-generic (a dense Gδ) subset of all the 2-dimensional conservative nonautonomous linear differential systems, either Lyapunov exponents are zero or there is a dominated splitting μ almost every point.
- Removing zero Lyapunov exponents in volume-preserving flowsPublication . Bessa, Mário; Rocha, JorgeBaraviera 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.
- Abundance of elliptic dynamics on conservative three-flowsPublication . Bessa, Mário; Duarte, PedroWe 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 Newhouse's 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.
- Generic dynamics of 4-dimensional C 2 hamiltonian systemsPublication . Bessa, Mário; Dias, João LopesWe 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 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].