# Géométrie et dynamique dans les espaces de modules

Séminaire Mensuel

## Séminaire - Juin

Le séminaire aura lieu (exceptionnellement) le mercredi 07/06 et le mercredi 14/06 (deux exposés) à l'Institut Henri Poincaré à Paris.  Pour télécharger les deux affiches du mois: juin1.pdf et juin2.pdf .

• 07/06/2023Jialun Li (École Polytechnique)  -  salle 05 de 14h à 15h

• Titre: On the Hausdorff dimension of the Rauzy gasket via stationary measures.

Resumé: I will discuss the proof of the equality between the Hausdorff dimension of the Rauzy gasket and the affinity dimension, based on joint work with Wenyu Pan and Disheng Xu. For the upper bound, we use a similar idea as in Sullivan's proof for conformal case $$SL(2,\mathbb{C})$$. For the lower bound, we prove the supremum of Hausdorff dimensions of stationary measures on the Rauzy gasket is no less than the affinity dimension through the following result. Let $$\nu$$ be a probability measure on $$SL(3,\mathbb{R})$$ whose support is finite and spans a Zariski dense subgroup. Let $$\mu$$ be the associated stationary measure for the action on the real projective plane. Under the exponential separation condition on $$\nu$$, we prove that the Hausdorff dimension of $$\mu$$ equals its Lyapunov dimension.

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• 14/06/2023Jon Chaika (University of Utah)
•
salle Olga Ladyjenskaïa (ex-salle 01) de 14h à 15h

Titre: There is a weakly mixing billiard in a polygon.

Resumé: Consider a point mass traveling in a polygon. It travels in a straight line, with constant speed, until it hits a side, at which point it obeys the rules of elastic collision. The main result of this talk is that there is such a flow that is weakly mixing with respect to the natural 3 dimensional volume on the unit tangent bundle to the polygon. This strengthens an earlier result of Kerckhoff, Masur and Smillie who proved the analogous result for ergodicity. Like Kerckhoff, Masur and Smillie's result, this follows from a theorem about translation surfaces, in our case that for every translation surface, for almost every pair of directions, $$a,b$$ we have that $$(F_a^t$$ x $$F_b^t)$$ is ergodic with respect to the Lebesgue measure on the product of the surface with itself. This is joint work with Giovanni Forni.

• 14/06/2023Francisco Arana-Herrera (University of Maryland)
•
salle Olga Ladyjenskaïa (ex-salle 01) de 15h15 à 16h15

Titre: Algebraic v/s geometric complexity of simple closed curves.

Resumé: Motivated by open questions of Sarnak, we explore the relation between algebraic and geometric complexity of simple closed curves on surfaces. We introduce a conjecture on the homological complexity of long simple closed hyperbolic geodesics and proceed to discuss a more accessible problem regarding the action in cohomology of mapping class groups. We explain the relation between these questions and mixing limit theorems for the Kontsevich-Zorich cocycle. We discuss a general framework for upgrading limit theorems to mixing limit theorems for dynamical systems under mild hyperbolicity and ergodicity assumptions. Parts of this talk are joint work in progress with Pouya Honaryar and other parts are joint work in progress with Giovanni Forni.