Some geometrical models of chaotic dynamics

Abstract

The free motion of a particle on a surface of constant negative curvature (a pseudosphere) was one of the first models of chaotic motion. It became the prototype for the theory of hyperbolic systems developed by Bowen and Sinai. In these models, geometry suggests a symbolic coding which already exhibits fully chaotic behaviour. One can return to these models to seek possible manifestations of quantum chaos. Here the mathematical technique is harmonic analysis on hyperbolic space. Chaotic behaviour seems to appear both in the behaviour of individual eigenfunctions and in the sequence of spectral values.