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- A Note on Expansiveness and Hyperbolicity for Generic Geodesic FlowsPublication . Bessa, MarioIn this short note we contribute to the generic dynamics of geodesic flows associated to metrics on compact Riemannian manifolds of dimension ≥ 2. We prove that there exists a C2-residual subset R of metrics on a given compact Riemannian manifold such tha tif g∈R,then its associated geodesic flow φgt is expansive if and only if the closure of the set of periodic orbits of φgt is a uniformly hyperbolic set. For surfaces, we obtain a stronger statement: there exists a C2-residual R such that if g ∈ R, then its associated geodesic flow φgt is expansive if and only if φgt is an Anosov flow.
- The Lyapunov exponents of generic skew-product compact semiflowsPublication . Bessa, Mario; Carvalho, Glória
- The flowbox theorem for divergence-free Lipschitz vector fieldsPublication . Bessa, MarioIn this note, we prove the flowbox theorem for divergence-free Lipschitz vector fields.
- Uniform hyperbolicity revisited: index of periodic points and equidimensional cyclesPublication . Bessa, Mario; Rocha, Jorge; Varandas, PauloIn this paper, we revisit uniformly hyperbolic basic sets and the dom- ination of Oseledets splittings at periodic points. We prove that peri- odic points with simple Lyapunov spectrum are dense in non-trivial basic pieces of Cr-residual diffeomorphisms on three-dimensional manifolds (r & 1). In the case of the C1-topology, we can prove that either all periodic points of a hyperbolic basic piece for a diffeomor- phism f have simple spectrum C1 -robustly (in which case f has a finest dominated splitting into one-dimensional sub-bundles and all Lya- punov exponent functions of f are continuous in the weak∗ -topology) or it can be C1-approximated by an equidimensional cycle associated to periodic points with robust different signatures. The latter can be used as a mechanism to guarantee the coexistence of infinitely many periodic points with different signatures.
- On the periodic orbits, shadowing and strong transitivity of continuous flowsPublication . Bessa, Mario; Torres, Maria Joana; Varandas, PauloWe prove that chaotic flows (i.e. flows that satisfy the shadowing property and have a dense subset of periodic orbits) satisfy a reparametrized gluing orbit property similar to the one introduced in Bomfim and Varandas (2015). In particular, these are strongly transitive in balls of uniform radius. We also prove that the shadowing property for a flow and a generic time-t map, and having a dense subset of periodic orbits hold for a C0-Baire generic subset of Lipschitz vector fields, that generate continuous flows. Similar results also hold for C0-generic homeomorphisms and, in particular, we deduce that chain recurrent classes of C0-generic homeomorphisms have the gluing orbit property.
- Stable weakly shadowable volume-preserving systems are volume-hyperbolicPublication . Bessa, Mário; Lee, Manseob; Vaz, SandraWe prove that any C1-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.
- Stable weak shadowable symplectomorphisms are partially hyperbolicPublication . Bessa, Mario; Vaz, SandraLet M be a closed, symplectic connected Riemannian manifold 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.
- On shadowing and hyperbolicity for geodesic flows on surfacesPublication . Bessa, Mario; Dias, João Lopes; Torres, Maria JoanaWe prove that the geodesic flow on closed surfaces displays a hyperbolic set if the shadowing property holds C2-robustly on the metric. Similar results are obtained when considering even feeble properties like the weak shadowing and the specification properties. Despite the Hamiltonian nature of the geodesic flow, the arguments in the present paper differ completely from those used in Bessa et al. (2013) for Hamiltonian systems.
- Straighten out coordinates for volume-preserving actionsPublication . Bessa, Mario; Morais, PedroIn this short note we obtain a canonical form for commuting divergence-free vector fields.
- Sobolev homeomorphisms are dense in volume preserving automorphismsPublication . Azevedo, Assis; Azevedo, Davide; Bessa, Mario; Torres, Maria JoanaIn this paper we prove a weak version of Lusin’s theorem for the space of Sobolev [...]