Ergodic Theory and Dynamical Systems

Research Article

Billiards on almost integrable polyhedral surfaces

E. Gutkina1

a1 Columbia University, New York, USA and Max-Planck-Institute für Mathematik, Bonn, West Germany†


The phase space of the geodesic flow on an almost integrable polyhedral surface is foliated into a one-parameter family of invariant surfaces. The flow on a typical invariant surface is minimal. We associate with an almost integrable polyhedral surface its holonomy group which is a subgroup of the group of motions of the Euclidean plane. We show that if the holonomy group is discrete then the flow on an invariant surface is ergodic if and only if it is minimal.

(Received November 16 1983)

(Revised April 09 1984)