Mathematical Sciences: Ergodic Phenomena in Differential Dynamics

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9404437 Sinai The proposed research will concern three different topics. The first is the study of ergodic like properties of standard-like maps. The main difficulty is due to the fact that in these cases, stable and unstable foliations are never transversal to each other. The new ideas and methods are taken from the theory of hyperbolic billiards. In the second topic, we shall study quasi-linear equations with random initial data and random coefficients. These equations appear in some cosmological problems. The analysis of statistical properties lead to new very interesting problems in probability theory. The third topic deals with some mathematical problems in the theory of quantum chaos. This is a new direction in mathematical physics which can be very promising. In general, we shall deal with several basic problems in the theory of chaos. The first one can be considered the most important open problem in Hamiltonian dynamics and concerns the appearance of chaotic behavior from pure deterministic laws of motion. The second one deals with equations of continuous media where the randomness is incorporated through initial data. It has important applications in cosmology. The last one concerns a new and difficult topic called quantum chaos and is of big interest for physics and mathematics. ***