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- Hamiltonian elliptic dynamics on symplectic $4$-manifoldsPublication . Bessa, Mário; Lopes Dias, JoãoWe consider C2-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 C2-generic Hamiltonian, the elliptic closed orbits are generic.
- Non-uniform hyperbolicity for infinite dimensional cocyclesPublication . Bessa, Mário; Carvalho, MariaLet H be an infinite dimensional separable Hilbert space, X a compact Hausdorff space and f : X \rightarrow X a homeomorphism which preserves a Borel ergodic measure which is positive on non-empty open sets. We prove that the non-uniformly Anosov cocycles are C0-dense in the family of partially hyperbolic f,H-skew products with non-trivial unstable bundles.
- On the entropy of conservative flowsPublication . Bessa, Mário; Varandas, PauloWe 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.
- Tracing orbits on conservative mapsPublication . Bessa, MárioWe explore uniform hyperbolicity and its relation with the pseudo orbit tracing property. This property indicates that a sequence of points which is nearly an orbit (affected with a certain error) may be shadowed by a true orbit of the system. We obtain that, when a conservative map has the shadowing property and, moreover, all the conservative maps in a C1-small neighborhood display the same property, then the map is globally hyperbolic
- Fine properties of Lp-cocycles which allow abundance of simple and trivial spectrumPublication . Bessa, Mário; Vilarinho, HelderIn this paper we prove that the class of accessible and saddle-conservative cocycles (a wide class which includes cocycles evolving in GL(d,R), SL(d,R) and Sp(d,R)) Lp-densely have a simple spectrum. We also prove that for an Lp-residual subset of accessible cocycles we have a one-point spectrum. Finally, we show that the linear differential system versions of previous results also hold and give some applications.
- 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.
- Generic hamiltonian dynamicsPublication . Bessa, Mário; Ferreira, Célia; Rocha, Jorge; Varandas, PauloIn this paper we contribute to the generic theory of Hamiltonians by proving that there is a C2-residual R in the set of C2 Hamiltonians on a closed symplectic manifold M, such that, for any H ∈ R, there is a full measure subset of energies e in H(M) such that the Hamiltonian level (H, e) is topologically mixing; moreover these level sets are homoclinic classes.
- Generic area-preserving reversible diffeomorphismsPublication . Bessa, Mário; Carvalho, Maria; Rodrigues, Alexandre A. P.Let M be a surface and R : M → M an area-preserving C∞ diffeomorphism which is an involution and whose set of fixed points is a submanifold with dimension one. We will prove that C1 -generically either an area-preserving R-reversible diffeomorphism, is Anosov, or, for μ-almost every x ∈ M, the Lyapunov exponents at x vanish or else the orbit of x belongs to a compact hyperbolic set with an empty interior. We will also describe a nonempty C1- open subset of area-preserving R-reversible diffeomorphisms where for C1-generically each map is either Anosov or its Lyapunov exponents vanish from almost everywhere.
- 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.
- On the Lyapunov spectrum of relative transfer operatorsPublication . Bessa, Mário; Stadlbauer, ManuelWe 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.