Expansion of the ergodic average of the horocycle flow

schedule le mardi 28 novembre 2017 de 10h30 à 12h00

Organisé par : D. Burguet, P-A. Guihéneuf

Intervenant : Alexander Adam (IMJ-PRG)
Lieu : Salle Paul Lévy, Campus Jussieu (salle 113, Tour 16/26)

Sujet : Expansion of the ergodic average of the horocycle flow

Résumé :
To an Anosov flow on a compact Riemannian manifold without boundary of dimension 3, one associates a special flow which has its flow direction pointing into the stable direction induced by the Anosov flow. 
This special flow is the horocycle flow. On relatively mild assumptions one knows that this horocycle flow is uniquely ergodic. 
A follow-up question is then how fast is the convergence to the Birkhoff average? 
In a paper of Livio Flaminio and Giovanni Forni (2003) they investigated this question for the geodesic flow on the unit tangent bundele of hyperbolic compact Riemann surfaces. 
They found that the speed of convergence is controlled by the existence of invariant distributions under the horocycle flow which are at the time also eigendistributions with certain eigenvalues for the geodesic flow. 
The speed of convergence is then determined by a power spectrum with exponents associated to those eigenvalues. 

I report on my study of this phenomenon for contact Anosov flows of sufficient regularity.