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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.
- A remark on the topological stability of symplectomorphismsPublication . Bessa, Mário; Rocha, JorgeWe prove that the C^1 interior of the set of all topologically stable C1 symplectomorphisms is contained in the set of Anosov symplectomorphisms.
- 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.
- Denseness of ergodicity for a class of volume-preserving flowsPublication . Bessa, Mário; Rocha, JorgeWe consider the class of C1 partially hyperbolic volume-preserving flows with one-dimensional central direction endowed with the C 1 -Whitney topology. We prove that, within this class, any flow can be approximated by an ergodic C2 volume-preserving flow and so, as a consequence, ergodicity is dense.
- 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.
- Topological stability for conservative systemsPublication . Bessa, Mário; Rocha, JorgeWe prove that the C 1 interior of the set of all topologically stable C1 incompressible flows is contained in the set of Anosov incompressible flows. Moreover, we obtain an analogous result for the discrete-time case.
- 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.
- Uniform hyperbolicity revisited: index of periodic points and equidimensional cyclesPublication . Bessa, Mário; Rocha, Jorge; Varandas, PauloIn this paper, we revisit uniformly hyperbolic basic sets and the domination of Oseledets splittings at periodic points. We prove that periodic points with simple Lyapunov spectrum are dense in non-trivial basic pieces of Cr-residual diffeomorphisms on three-dimensional manifolds (r>=1). In the case of the C1-topology, we can prove that either all periodic points of a hyperbolic basic piece for a diffeomorphism f have simple spectrum C1 -robustly (in which case f has a finest dominated splitting into one-dimensional sub-bundles and all Lyapunov exponent functions of f are continuous in the weak∗ -topology) or it can be C1-approximated by an equidimensional cycle associated to periodic points with robust different signatures. The latter can be used as a mechanism to guarantee the coexistence of infinitely many periodic points with different signatures.
- Estabilidade de hamiltonianosPublication . Bessa, Mário; Rocha, Jorge; Torres, Maria JoanaNesta breve nota considera-se o contexto dos sistemas Hamiltonianos, definidos numa variedade simplética M de dimensão 2d (d >= 2). Prova-se que um sistema Hamiltoniano estrela é Anosov. Como consequência obtém-se a prova da conjetura da estabilidade para Hamiltonianos. Prova-se ainda que um sistema Hamiltoniano H é Anosov se qualquer das seguintes afirmações se verifica: H é robustamente topologicamente estável; H é estavelmente sombreável; H é estavelmente expansivo; e H possui a propriedade de especificação fraca estável. Além disso, para um Hamiltoniano C2-genérico, a união das hipersuperfícies de energia regulares parcialmente hiperbólicas e das órbitas fechadas elípticas, forma um subconjunto denso de M. Como consequência, qualquer hipersuperfície de energia regular robustamente transitiva de um Hamiltoniano C2 é parcialmente hiperbólica. Por fim, as hipersuperfícies de energia regulares estavelmente fracamente sombreáveis são parcialmente hiperbólicas.
- Contributions to the geometric and ergodic theory of conservative flowsPublication . Bessa, Mário; Rocha, JorgeWe prove the following dichotomy for vector fields in a C1-residual subset of volume-preserving flows: for Lebesgue-almost every point, either all of its Lyapunov exponents are equal to zero or its orbit has a dominated splitting. Moreover, we prove that a volume-preserving and C1-stably ergodic flow can be C1-approximated by another volume-preserving flow which is non-uniformly hyperbolic.