Hyperbolicity and astigmatism

We study the mechanism of hyperbolicity in high-dimensional Hamiltonian sys tems. Especially we consider ergodic billiards with focusing components in dimensions; d greater than or equal to 3. In this case astigmatism serves a s an obstacle to hyperbolicity in billiards with large focusing components. The notion of absolutely focusing mirrors is extended to the dimensions d greater than or equal to 3 and the first classes of ergodic billiards with both focusing and dispersing components are constructed in d greater than o r equal to 3.