100, rue des maths 38610 Gières / GPS : 45.193055, 5.772076 / Directeur : Thierry Gallay

Mario Shannon

Birkhoff sections for 3-dimensional flows and the problem of conjugacy
Vendredi, 1 Février, 2019 - 10:30
Résumé : 

Let M be a closed 3-manifold and consider a flow generated by a vector field X in M. A Birkhoff section for this flows is an embedding of a compact surface S in M satisfying that the interior of S embeds transverse to the flow lines, the boundary components of S are mapped onto periodic orbits of the flow, and every orbit of the flow intersect S in an uniformly bounded time. When the boundary of S is empty then S is just a global section for the flow. It follows that there exists a well-defined first-return map to the interior of S and, in the complement of the boundary, the flow is a suspension flow up to topological equivalence,.

Given two flows and one Birkhoff section for each flow, assume that the first-return maps are conjugated as homeomorphisms. We ask under which conditions this implies that the flows are actually topologically conjugated. We show that, under the hypothesis that the boundary of S consists of saddle-type hyperbolic periodic orbits plus some topological conditions, the answer is affirmative. 
The aim of the talk is to show some ideas involved in the proof of this property, as well as some application to the study of transitive Anosov flows.    
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