6533b820fe1ef96bd127a136

RESEARCH PRODUCT

Dehn surgeries and smooth structures on 3-dimensional transitive Anosov flows.

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The present thesis is about Dehn surgeries and smooth structures associated with transitive Anosov flows in dimension three. Anosov flows constitute a very important class of dynamical systems, because of its persistent chaotic behaviour, as well as for its rich interaction with the topology of the ambient space. Even if a lot is known about the dynamical and ergodic properties of these systems, there is not a clear understanding about how to classify its different orbital equivalence classes. Until now, the biggest progress has been done in dimension three, where there is a family of techniques intended for the construction of Anosov flows called surgeries.During the realization of this thesis, in a first time we have been interested in a particular surgery method, known as the Goodman surgery. This method consists in make a Dehn surgery on a chosen periodic orbit, but adapted to the flow, in such a way to obtain a new manifold equipped with an Anosov flow. For making this surgery, one of the parameters that has to be chosen is an embedded surface in the 3-manifold and a diffeomorphism defined on it. Thus, the parameter space is, a priori, of infinite dimension and it is not easy to have control on the orbital equivalence class of the obtained flow. There exists a second method, that can be interpreted as an infinitesimal version of the previous one, known as the Fried surgery. It consists in making a blow-up of the flow along the periodic orbit, obtaining in this way a flow in a manifold with boundary, for then blowing-down the boundary component in a non-trivial way and produce a new flow. This surgery produces flows defined in a unique way, but they are not equipped with a natural uniformly hyperbolic structure. They are, by construction, topological Anosov flows.Our contribution is to show that, if we assume that the flow is transitive, then a Goodman surgery or a Fried surgery performed on a periodic orbit produce orbitally equivalent flows, for the same choice of integer parameters.In a second time, we have been interested for a more abstract question, but which is also related to some technical issues in the construction of hyperbolic flows. It is the problem of determining if every topologically Anosov flow (i.e. expansive and satisfying the Bowen shadowing property) correspond to a smooth hyperbolic flow, up to orbital equivalence. In the particular case that the flow is transitive, it has been known that there exists a non-uniformly hyperbolic structure defined in the complement of a finite set of periodic orbits. The main difficulty is the construction of (global) hyperbolic models associated to the original flow.In this setting, our contribution is to show that every transitive topologically Anosov flow on a closed manifold is orbital equivalent to a smooth Anosov flow.