Chaotic Dynamics, Attractors and Repellers

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AbstractChernov The project is devoted to mathematical problems in equilibrium and nonequilibrium statistical mechanics, focusing on the ergodic and statistical properties of deterministic chaotic dynamical systems. Specific models include Lorentz gases in external fields, viscous gases of hard balls, open Hamiltonian systems with escape holes. We are going to study rigorously several laws of physics (including Ohm's law, Green-Kubo formula and Einstein relation) for multiparticle Lorentz gases in external fields with discrete and continuous times. We expect to prove exponential decay of correlations, Bernoulli property and various probabilistic limit theorems. The ergodic properties and entropy of hard ball gases will also be studied. Another direction of research is to investigate conditionally invariant measures, escape-rate formalism, and fractal structure of chaotic repellers in open systems (with escape holes). The mathematical theory of chaos and dynamical systems has grown dramatically over the past 40 years. It started with foundations of statistical physics created by Boltzmann and Maxwell. It has covered basic biological models (dynamics of population), various problems in engineering, earthquake prediction, traffic control, neural networks for computer data processing, optimization of manufacturing and management tasks, and has countless other applications where long-time behavior of complex systems is in question. Universal features of chaos exhibited by various processes in nature and society make it possible to describe and predict such phenomena by the means of one theory. In particular, the concepts of attractors and repellers (as structures or patterns determining long-time evolution of complex systems) are basic categories in the theory of chaos. Mathematical theory of chaotic attractors and repellers is still relatively young and quite different from the old, classical mathematics.