Nonuniformly Hyperbolic Dynamics

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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

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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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Non-uniform hyperbolicity for infinite dimensional cocycles

Publication . Bessa, Mário; Carvalho, Maria

Let 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.

2013-07-18Journal article

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On C1-generic chaotic systems in three-manifolds

Publication . Bessa, Mário

Let M be a closed 3-dimensional Riemannian manifold. We exhibit a C1-residual subset of the set of volume-preserving 3-dimensional flows defined on very general manifolds M such that, any flow in this residual has zero metric entropy, has zero Lyapunov exponents and, nevertheless, is strongly chaotic in Devaney’s sense. Moreover, we also prove a corresponding version for the discrete-time case.

2013Journal article

Open access