Géométrie et dynamique dans les espaces de modules

Séminaire Mensuel

Séminaire - Septembre

Le séminaire aura lieu deux mercredis ce mois (exceptionnellement) de 14h à 15h à l'Institut Henri Poincaré à Paris.  Pour télécharger le programme du mois: septembre.pdf.

  • 20/09/2017 - Anatole Katok (Penn State University)  -  salle 05

  • Titre: Around surfaces of negative curvature.

    Resumé: I will discuss relations (or the absence of those) between global geometric, dynamical and conformal characteristics. The recent entropy flexibility result joint with A. Erchenko states that for any integer \(G >1\) and positive \(V\) and any two numbers \(A,B\) such that \(0 < A < ( 4\pi(G - 1)/V)^{1/2} < B\) there exists a Riemannian metric of negative curvature on the surface of genus \(G\) with total area \(V\) such that the entropy of the geodesic flow with respect to the Liouville measure is equal to \(A\) and its topological entropy is equal to \(B\). An interesting open question is whether this is also true within a fixed conformal class of metrics. A necessary condition is existence within conformal class of metrics of negative curvature with bounded diameter and arbitrary short systole. Further global invariants include the average of the square root of the absolute value of the curvature, entropy with respect to the harmonic measure, positive Lyapunov exponent with respect to the measure of maximal entropy, and the global conformal coefficient. A strong flexibility conjecture states that aside from known basic inequalities there are restrictions on the values of those quantities. While the first of the those quantities can be brought into the picture by methods similar to those used in the proof of entropy flexibility, the others present challenging problems.

  • 27/09/2017 - Rodolfo Gutierrez (IMJ-PRG)  -  salle 05

  • Titre: Zariski density of Rauzy–Veech groups

    Resumé: The Kontsevich–Zorich conjecture, stating the simplicity of the Lyapunov spectra of almost any interval exchange transformation or translation flow with respect to the Masur–Veech measures, was solved by Avila and Viana by showing that the Rauzy–Veech groups are pinching and twisting. Their solution explicitly avoids computing the Zariski closure of such groups, which was conjectured by Zorich to be the entire ambient symplectic group. Recently, Avila, Yoccoz and Matheus proved that the hyperelliptic Rauzy–Veech groups are finite-index subgroups of the symplectic group, which implies that they are Zariski-dense and, thus, solves the conjecture for the hyperelliptic case. In this talk I will present a proof of the finite-indexness of every Rauzy–Veech group. The proof relies on explicitly computing the Rauzy–Veech groups for the non-hyperelliptic components of minimal strata (that is, strata with a single zero) and then using a variant of the combinatorial adjacency of strata found by Avila and Viana to reduce the general case to the minimal one.