Limiting distributions of smooth curves under geodesic flow on hyperbolic manifolds

Jeudi 29 mai 2008 14:00-15:00 - Shah Nimish - Tata Institute, Bombay

Résumé : We study the asymptotic behaviour of the evolution of a compact piece of a C^n-curve under the geodesic flow on the unit tangent bundle of a compact (or finite volume) hyperbolic n-manifold.
We show that under a natural geometric condition on the curve, the normalized parameter measure on the curve when pushed foward under the geodesic flow gets equidistributed with respect to the natural normalized measure on the unit tangent bundle ; in particular, the translated curve starts getting dense in the space.
The condition can be described as follows : Under the visual map, any lift of (a set of positive parameter measure of) the curve to the covering space does not project on any proper subsphere of the boundary sphere. For example, if the curve is analytic, under the visual map any of its lift should not map to a proper subsphere of the boundary sphere. In fact, we only need to avoid those subspheres whose bounding hyperbolic disk (in the covering space) projects onto a compact (or closed) totally geodesic immersed submanifold.

Lieu : bât. 425 - 121-123

Limiting distributions of smooth curves under geodesic flow on hyperbolic manifolds  Version PDF