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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.
- Billiards in generic convex bodies have positive topological entropyPublication . Bessa, Mário; Del Magno, Gianluigi; Dias, João Lopes; Gaivão, José Pedro; Torres, Maria JoanaWe show that there exists a C2-open dense set of convex bodies with smooth boundary whose billiard map exhibits a non-trivial hyperbolic basic set. As a consequence billiards in generic convex bodies have positive topological entropy and exponential growth of the number of periodic orbits.
- Expansiveness and hyperbolicity in convex billiardsPublication . Bessa, Mário; Dias, João Lopes; Torres, Maria JoanaWe say that a convex planar billiard table B is C2-stably expansive on a fixed open subset U of the phase space if its billiard map f_B is expansive on the maximal invariant set Λ_{B,U}, and this property holds under C2-perturbations of the billiard table. In this note we prove for such billiards that the closure of the set of periodic points of f_B in Λ_{B,U} is uniformly hyperbolic. In addition, we show that this property also holds for a generic choice among billiards which are expansive.
- Hamiltonian suspension of perturbed Poincaré sections and an applicationPublication . Bessa, Mário; Dias, João LopesWe 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.
- Generic hamiltonian dynamical systems: an overviewPublication . Bessa, Mário; Dias, João LopesWe present for a general audience the state of the art on the generic properties of C 2 Hamiltonian dynamical systems.
- On shadowing and hyperbolicity for geodesic flows on surfacesPublication . Bessa, Mário; 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.
- Generic dynamics of 4-dimensional C 2 hamiltonian systemsPublication . Bessa, Mário; Dias, João LopesWe study the dynamical behaviour of Hamiltonian flows defined on 4-dimensional compact symplectic manifolds. We find the existence of a C 2-residual set of Hamiltonians for which there is an open mod 0 dense set of regular energy surfaces each either being Anosov or having zero Lyapunov exponents almost everywhere. This is in the spirit of the Bochi-Mañé dichotomy for area-preserving diffeomorphisms on compact surfaces [2] and its continuous-time version for 3-dimensional volume-preserving flows [1].
- Hyperbolicity through stable shadowing for generic geodesic flowsPublication . Bessa, Mário; Dias, João Lopes; Torres, Maria JoanaWe prove that the closure of the closed orbits of a generic geodesic flow on a closed Riemannian n ≥ 2 dimensional manifold is a uniformly hyperbolic set if the shadowing property holds C2-robustly on the metric. We obtain analogous results using weak specification and the shadowing property allowing bounded time reparametrization.