## 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
- 27/09/2017 - Rodolfo Gutierrez (IMJ-PRG) - 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.

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.