Billiard-Type Systems with Chaotic Behaviour and Space-Time Chaos

In the last two decades the subject of the branch of mathematics that is called Dynamical Systems is almost identified with the study of chaotic motion of such systems. However until recently there were very few examples of in a sense realistic systems that have been proven to be chaotic. The first found models of that type were geodesic flows on manifolds of negative curvature (see [Ha], [He], [H1]). The general ideas of these papers were developed by D. V. Anosov, Ya. G. Sinai and S. Smale (see [An], [AS], [S2], [Sm]) and lead to the concept of hyperbolicity as the basic mechanism of chaos in classical dynamical systems. This concept together with the closely related notion of Lyapunov exponents (see [O], [P]) serve as the foundation of the modern theory of nonuniformly hyperbolic dynamical systems.