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- 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.
- 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.