Strategic Project - UI 212 - 2011-2012

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Publications

Stable weak shadowable symplectomorphisms are partially hyperbolic

Publication . Bessa, Mário; Vaz, Sandra

Let M be a closed, symplectic connected Riemannian mani- fold and f a symplectomorphism on M. We prove that if f is C1-stably weak shadowable on M, then the whole manifold M admits a partially hyperbolic splitting.

2014Journal article

Open access

Fine properties of Lp-cocycles which allow abundance of simple and trivial spectrum

Publication . Bessa, Mário; Vilarinho, Helder

In 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.

2014Journal article

Open access

Hamiltonian suspension of perturbed Poincaré sections and an application

Publication . Bessa, Mário; Dias, João Lopes

We construct a Hamiltonian suspension for a given symplectomorphism which is the per- turbation of a Poincaré map. This is especially useful for the conversion of perturbative results between symplectomorphisms and Hamiltonian flows in any dimension 2d. As an application, using known properties of area-preserving maps, we prove that for any Hamiltonian defined on a symplectic 4-manifold M and any point p ∈ M, there exists a C2-close Hamiltonian whose regular energy surface through p is either Anosov or contains a homo- clinic tangency.

2014Journal article

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Shades of hyperbolicity for hamiltonians

Publication . Bessa, Mário; Rocha, Jorge; Torres, Maria Joana

We prove that a Hamiltonian system H ∈ C2(M,R) is globally hyperbolic if any of the following statements hold: H is robustly topologically stable; H is stably shadowable; H is stably expansive; and H has the stable weak specification property. Moreover, we prove that, for a C2-generic Hamiltonian H , the union of the partially hyperbolic regular energy hypersurfaces and the closed elliptic orbits, forms a dense subset of M. As a consequence, any robustly transitive regular energy hypersurface of a C2-Hamiltonian is partially hyperbolic. Finally, we prove that stable weakly-shadowable regular energy hypersurfaces are partially hyperbolic.

2013Journal article

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Generic area-preserving reversible diffeomorphisms

Publication . 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.

2015Journal article

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