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Billiards in generic convex bodies have positive topological entropy

Publication . Bessa, Mário; Del Magno, Gianluigi; Dias, João Lopes; Gaivão, José Pedro; Torres, Maria Joana

We 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.

2024Journal article

Open access

Topological aspects of incompressible flows

Publication . Bessa, Mário; Torres, Maria Joana; Varandas, Paulo

In 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.

2021Journal article

Restricted access

Hyperbolicity through stable shadowing for generic geodesic flows

Publication . Bessa, Mário; Dias, João Lopes; Torres, Maria Joana

We 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.

2020Journal article

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The closing lemma and the planar general density theorem for Sobolev maps

Publication . Azevedo, Assis; Azevedo, Davide; Bessa, Mário; Torres, Maria Joana

We 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.

2021Journal article

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The role of the saddle-foci on the structure of a bykov attracting set

Publication . Bessa, Mário; Carvalho, Maria; Rodrigues, Alexandre A. P.

We consider a one-parameter family ( fλ)λ 􏰅 0 of symmetric vector fields on the three-dimensional sphere whose flows exhibit a heteroclinic network between two saddle-foci inside a global attracting set. More precisely, when λ = 0, there is an attracting heteroclinic cycle between the two equilibria which is made of two 1- dimensional connections together with a 2-dimensional sphere which is both the stable manifold of one saddle-focus and the unstable manifold of the other. After slightly increasing the parameter while keeping the 1-dimensional connections unaltered, the two-dimensional invariant manifolds of the equilibria become transversal, and thereby create homoclinic and heteroclinic tangles. It is known that these newborn structures are the source of a countable union of topological horseshoes, which prompt the coexistence of infinitely many sinks and saddle-type invariant sets for many values of λ. We show that, for every small enough positive parameter λ, the stable and unstable manifolds of the saddle-foci and those infinitely many horseshoes are contained in the global attracting set of fλ; moreover, the horseshoes belong to the heteroclinic class of the equilibria. In addition, we show that the set of chain-accessible points from either of the saddle-foci is chain-stable and contains the closure of the invariant manifolds of the two equilibria.

2020Journal article

Open access