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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.
- 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.
- 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.
- 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.
- 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.
- Three-dimensional conservative star flows are AnosovPublication . Bessa, Mário; Rocha, JorgeA divergence-free vector field satisfies the star property if any divergence-free vector field in some C1-neighborhood has all the singularities and all closed orbits hyperbolic. In this article we prove that any divergence- free star vector field defined in a closed three-dimensional manifold is Anosov. Moreover, we prove that a C1-structurally stable three-dimensional conserva- tive flow is Anosov.
- 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.
- Hyperbolicity and stability for Hamiltonian flowsPublication . Bessa, Mário; Rocha, Jorge; Torres, Maria JoanaWe prove that a Hamiltonian star system, defined on a 2d-dimen- sional symplectic manifold M (d 2), is Anosov. As a consequence we obtain the proof of the stability conjecture for Hamiltonians. This generalizes the 4-dimensional results in Bessa et al. (2010) [5].
- Stable weakly shadowable volume-preserving systems are volume-hyperbolicPublication . Bessa, Mário; Lee, Manseob; Vaz, SandraWe prove that any C^1-stable weakly shadowable volume-preserving diffeomorphism defined on a compact manifold displays a dominated splitting E ⊕ F . Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result for divergence-free vector fields. As a consequence, in low dimensions, we obtain global hyperbolicity.
- On the Lyapunov spectrum of infinite dimensional random products of compact operatorsPublication . Bessa, Mário; Carvalho, MariaWe consider an infinite dimensional separable Hilbert space and its family of compact integrable cocycles over a dynamical system f. Assuming that f acts in a compact Hausdorff space X and preserves a Borel regular ergodic probability which is positive on non-empty open sets, we conclude that there is a C0-residual subset of cocycles within which, for almost every x, either the Oseledets–Ruelle’s decomposition along the orbit of x is dominated or all the Lyapunov exponents are equal to −∞.