Hyperbolicity and astigmatism

摘要

We study the mechanism of hyperbolicity in high-dimensional Hamiltonian systems. Especially we consider ergodic billiards with focusing components in dimensions; d greater than or equal to 3. In this case astigmatism serves as 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 or equal to 3. [References: 17]