ABUNDÂNCIA DE EXPOENTES DE IYAPUNOV ZERO EM SISTEMAS CONSERVATIVOS A TEMPO CONTÍNUO

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Topological stability for conservative systems

Publication . Bessa, Mário; Rocha, Jorge

We 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.

2011Journal article

Open access

On the entropy of conservative flows

Publication . Bessa, Mário; Varandas, Paulo

We obtain a C1-generic subset of the incompressible flows in a closed three-dimensional manifold where Pesin’s entropy formula holds thus establishing the continuous-time version of Tahzibi (C R Acad Sci Paris I 335:1057–1062, 2002). Moreover, in any compact manifold of dimension larger or equal to three we obtain that the metric entropy function and the integrated upper Lyapunov exponent function are not continuous with respect to the C1 Whitney topology. Finally, we establish the C2- genericity of Pesin’s entropy formula in the context of Hamiltonian four-dimensional flows.

2010Journal article

Open access

Removing zero Lyapunov exponents in volume-preserving flows

Publication . Bessa, Mário; Rocha, Jorge

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.

2007Journal article

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

Publication . Bessa, Mário; Ferreira, Celia; Rocha, Jorge

A Hamiltonian level, say a pair $(H,e)$ of a Hamiltonian $H$ and an energy $e \in \mathbb{R}$, is said to be Anosov if there exists a connected component $\mathcal{E}_{H,e}$ of $H^{-1}({e})$ which is uniformly hyperbolic for the Hamiltonian flow $X_H^t$. The pair $(H,e)$ is said to be a Hamiltonian star system if there exists a connected component $\mathcal{E}^\star_{H,e}$ of the energy level $H^{-1}({{e}})$ such that all the closed orbits and all the critical points of $\mathcal{E}^\star_{H,e}$ are hyperbolic, and the same holds for a connected component of the energy level $\tilde{H}^{-1}({\tilde{e}})$, close to $\mathcal{E}^\star_{H,e}$, for any Hamiltonian $\tilde{H}$, in some $C^2$-neighbourhood of $H$, and $\tilde{e}$ in some neighbourhood of $e$. In this article we prove that for any four-dimensional Hamiltonian star level $(H,e)$ if the surface $\mathcal{E}^\star_{H,e}$ does not contain critical points, then $X_H^t|_{\mathcal{E}^\star_{H,e}}$ is Anosov; if $\mathcal{E}^\star_{H,e}$ has critical points, then there exists $\tilde{e}$, arbitrarily close to $e$, such that $X_H^t|_{\mathcal{E}^\star_{H,\tilde{e}}}$ is Anosov.

2009-09-21Journal article

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Dominated splitting and zero volume for incompressible three flows

Publication . Araujo, Vitor; Bessa, Mário

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 Poincaré flow, then it must be an Anosov flow. With this result we are able to extend the dichotomies of Bochi–Mañé (see Bessa 2007 Ergod. Theory Dyn. Syst. 27 1445–72, Bochi 2002 Ergod. Theory Dyn. Syst. 22 1667–96, Mañé1996 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.

2008Journal article

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