Loading...
13 results
Search Results
Now showing 1 - 10 of 13
- Topological aspects of incompressible flowsPublication . Bessa, Mário; Torres, Maria Joana; Varandas, PauloIn this article we approach some of the basic questions in topological dynamics, concerning periodic points, transitivity, the shadowing and the gluing orbit properties, in the context of C0 incompressible flows generated by Lipschitz vector fields. We prove that a C0-generic incompressible and fixed-point free flow satisfies the periodic shadowing property, it is transitive and has a dense set of periodic points in the non- wandering set. In particular, a C0-generic fixed-point free incompressible flow satisfies the reparametrized gluing orbit property. We also prove that C0-generic incompressible flows satisfy the general density theorem and the weak shadowing property, moreover these are transitive.
- Sobolev homeomorphisms are dense in volume preserving automorphismsPublication . Azevedo, Assis; Azevedo, Davide; Bessa, Mário; Torres, Maria JoanaIn this paper we prove a weak version of Lusin’s theorem for the space of Sobolev-(1,p) volume preserving homeomor- phisms on closed and connected n-dimensional manifolds, n ≥ 3, for p < n − 1. We also prove that if p > n this result is not true. More precisely, we obtain the density of Sobolev-(1,p) homeomorphisms in the space of volume pre- serving automorphisms, for the weak topology. Furthermore, the regularization of an automorphism in a uniform ball cen- tered at the identity can be done in a Sobolev-(1, p) ball with the same radius centered at the identity.
- 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.
- The closing lemma and the planar general density theorem for Sobolev mapsPublication . Azevedo, Assis; Azevedo, Davide; Bessa, Mário; Torres, Maria JoanaWe prove that given a non-wandering point of a Sobolev-(1,p) homeomorphism we can create closed trajectories by making arbitrarily small perturbations. As an application, in the planar case, we obtain that generically the closed trajectories are dense in the non-wandering set.
- 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.
- The C0 general density theorem for geodesic flowsPublication . Bessa, Mário; Torres, Maria JoanaGiven a closed Riemannian manifold, we prove the C0-general density theorem for continuous geodesic flows. More precisely, we prove that there exists a residual (in the C0-sense) subset of the continuous geodesic flows such that, in that residual subset, the geodesic flow exhibits dense closed orbits.
- On the periodic orbits, shadowing and strong transitivity of continuous flowsPublication . Bessa, Mário; 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.
- 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.
- Estabilidade de hamiltonianosPublication . Bessa, Mário; Rocha, Jorge; Torres, Maria JoanaNesta breve nota considera-se o contexto dos sistemas Hamiltonianos, definidos numa variedade simplética M de dimensão 2d (d >= 2). Prova-se que um sistema Hamiltoniano estrela é Anosov. Como consequência obtém-se a prova da conjetura da estabilidade para Hamiltonianos. Prova-se ainda que um sistema Hamiltoniano H é Anosov se qualquer das seguintes afirmações se verifica: H é robustamente topologicamente estável; H é estavelmente sombreável; H é estavelmente expansivo; e H possui a propriedade de especificação fraca estável. Além disso, para um Hamiltoniano C2-genérico, a união das hipersuperfícies de energia regulares parcialmente hiperbólicas e das órbitas fechadas elípticas, forma um subconjunto denso de M. Como consequência, qualquer hipersuperfície de energia regular robustamente transitiva de um Hamiltoniano C2 é parcialmente hiperbólica. Por fim, as hipersuperfícies de energia regulares estavelmente fracamente sombreáveis são parcialmente hiperbólicas.
- 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].