5/3/2005

Omri Sarig
Penn State University

Ergodic theory for the horocycle flow on periodic hyperbolic surfaces (joint with F. Ledrappier)

The horocycle flow of a non-geometrically finite hyperbolic surface may have many different (infinite) ergodic invariant Radon measures. (This should be contrasted with co-compact or co-finite volume case where there is just one non-trivial measure - up to scaling). We classify these measures for the class of periodic surfaces: regular covers of surfaces of finite volume. It turns out that in this case there are as many measures as there are positive eigenfunctions for the Laplacian of the surface. In some situations, only one of these measures is "relevant" from an ergodic theoretic point of view.