Séminaire Géométrie Topologie Dynamique
Rigidity for conjugacies of dissipative Anosov flows in dimension 3
June 2024
Intervenant : Martin Leguil
Institution : Polytechnique
Heure : 14h00 - 15h00
Lieu : 2L8

In an ongoing project with A. Gogolev and F. Rodriguez Hertz, we study when two transitive Anosov flows in dimension 3 which are topologically conjugated are actually smoothly conjugated. By the work of de la Llave, Marco and Moriyón from the 80s, a necessary and sufficient condition for that is that stable and unstable eigenvalues at corresponding periodic points match. In our work we show that in many cases, the latter condition is redundant, as it is already implied by the existence of a topological conjugacy. In particular, we show that the conjugacy is smooth, unless one of the flows is a suspension, or the conjugacy swaps positive and negative SRB measures of the two flows. This extends a recent work of Gogolev-Rodriguez Hertz in the volume preserving case. I will try to explain how this rigidity problem is connected with other notions, in particular, the Foulon-Hasselblatt cocycle, and the so-called templates introduced by Tsujii and Zhang to study the regularity of stable and unstable distributions.

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