Martin Leguil (Université de Picardie)

Title:
Spectral Rigidity of hyperbolic billiards

Abstract:
In a project with P. Bálint, J. De Simoi and V. Kaloshin, we have studied the inverse problem for a class of open dispersing billiards satisfying a non-eclipse condition. The dynamics of such billiards is of type Axiom A and can be coded symbolically in a natural way, which allows us to define a marked length spectrum (lengths of all periodic orbits + coding). We have shown that this dynamical spectrum contains a lot of geometric information; in particular, when the billiard table has analytic boundary and a partial Z_2\times Z_2 symmetry, it is generically possible to determine the geometry of the table from its marked length spectrum. 

In a joint work with A. Florio, we have carried on with the study of such billiards in the case where the boundary is merely C^k. We show that open dispersing billiards satisfying the non-eclipse condition are spectrally rigid in the following sense: two such billiards which have the same marked length spectrum share the same geometry at the points of the table « seen » by periodic orbits, i.e., on the projection of the Cantor set of trapped orbits on the boundary of the table. In particular, when the boundary of these tables is (quasi-)analytic, this implies that the two tables are isometric. One step of the proof is a general dynamical result on smooth conjugacy classes of 3D contact Axiom A flows, which extends a previous result due to J. Feldman and D. Ornstein in the Anosov case. Some ideas of our work are inspired by the techniques introduced by J.-P. Otal in his work on the spectral rigidity of geodesic flows on negatively curved surfaces.