Hyperbolicity and Astigmatism
by
Leonid A. Bunimovich
Georgia Institute of Technology

Hyperbolicity is the origin of stochasticity of classical dynamical systems. The geodesic flows on surfaces of a constant negative curvature were the first such systems proven to be chaotic (Hadamard, Hedlund, Hopf). This approach has been essentially generalized in the studies of general uniformly hyperbolic systems (Anosov, Sinai, Smale).

All those systems are smooth though while the classical models in physics (e.g. a gasof hard spheres) do have singularities and therefore belong to the class of nonuniformly hyperbolic dynamical systems. The theory of such systems has been developed by Sinai, Pesin and others. Sinai has introduced the class of billiards with smooth convex inwards boundary which are nonuniformly hyperbolic systems and correspond in a sense to geodesic flows on manifolds of negative curvature. He proved that these billiards are ergodic and have strong chaotic properties.

On the other hand, the classical integrable geodesic flows on the surfaces of positive curvature together with the classical examples of integrable focusing billiards (circles and ellipses) seemed to demonstrate that focusing produced by a positive curvature or by a convex ourwards boundary always stabilize the dynamics. This intuition was strongly held by the mathematicians as well as by physicists.

Therefore the discovery of chaotic focusing billiards made in 1974 came to the complete surprise of both communuities. It demonstrated that there exists another mechanism of hyperbolicity besides dispersing. This mechanism has been called the mechanism of defocusing.

The same idea was applied to constuct ergodic geodesic flows on a two dimensional sphere and on two dimensional torus as well as on some other manifolds (Osserman, Donnay, Burns Gerber). However all these examples were essentially two-dimensional.

So the question whether or not the mechanism of defocusing may work in higher dimensions remained open since 1974. At first sight, the answer to this question should be negative. The reason for that is the well known phenomenon in optics, which is called an astigmatism. The astigmatism means that focusing in higher-dimensional spaces (d > 2) is not the same in different two-dimensional planes.

We will present the resent results on the existence of chaotic nowhere dispersing (i.e. focusing) billiards in all (finite) dimensions. These billiards are hyperbolic and Bernoulli.

Paper reference: doi:10.1023/A:1026405920274

Date received: May 6, 1999