Centre of Mathematics of the University of Minho

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Publications

Sobolev homeomorphisms are dense in volume preserving automorphisms

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

In 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.

2019Journal article

Open access

On shadowing and hyperbolicity for geodesic flows on surfaces

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

We 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.

2017Journal article

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On the periodic orbits, shadowing and strong transitivity of continuous flows

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

We 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.

2018Journal article

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There are no proper topological hyperbolic homoclinic classes for area-preserving maps

Publication . Bessa, Mário; Torres, Maria Joana

We begin by defining a homoclinic class for homeomorphisms. Then we prove that if a topological homoclinic class Λ associated with an area-preserving homeomorphism f on a surface M is topologically hyperbolic (i.e. has the shadowing and expansiveness properties), then Λ = M and f is an Anosov homeomorphism.

2019Journal article

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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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