Transition to Chaos in Classical and Quantum Mechanics: Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme, Italy, July 6 - 13, 1991 - 1589 (1994 edition.)

The Session was intended to give a broad survey of the mathematical problems arising in the chaotic transition of deterministic dynamical systems, both in classical and quantum mechanics.The lectures of Mather and Forni thoroughly cover the area- preserving twist maps, and include an up-to-date version of the Aubry-Mather theory.The lectures of Bellissard describe the quantum aspects: classical limit, localization, spectral properties of the relevant Schrodinger operators and thus represent an exhaus- tive introduction to the mathematics of "quantum chaos".

The lectures of Degli Esposti et al. reviews equidistribu- tion of unstable periodic orbits and classical limit of quantized toral symplectomorphisms.