Chaos, Quantization and the Classical Limit on the Torus

The algebraic and the canonical approaches to the quantization of a class of classical symplectic dynamical systems on the two-torus are presented in a simple unified framework. This allows for ready comparison between the two very different approaches and is well adapted to the study of the semi-classical behaviour of the resulting models. Ergodic translations and skew translations, as well as the hyperbolic toral automorphisms and their Hamiltonian perturbations are treated. Ergodicity is proved for the algebraic quantum model of the translations and skew-translations and exponential mixing in the algebraic quantum model of the hyperbolic automorphisms. This latter result is used to show the non-commutativity of the classical and large time limits. Turning to the canonical model, recent results are reviewed on the behaviour in the classical limit of the eigenvalues and eigenvectors of the quantum propagators; the link with the ergodic or mixing properties of the underlying dynamics is explained. An example of the non-commutativity of the classical and large-time limits is proven here as well.