Mathematical Sciences: Chaotic Attractors and Repellers: Rigorous Results

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Abstract Chernov The project is devoted to chaotic dynamical systems. It focuses on nonequilibrium statistical physics, chaotic attractors, and repellers. A major goal of this project is to find effective bounds on correlation functions for continuous-time hyperbolic dynamical systems with singularities, such as periodic Lorentz gas. This will make it possible to derive rigorously transport laws for related models of nonequilibrium thermodynamics. These include Ohm's law, the Einstein relation and the Green-Kubo formula for a charged particle in the periodic Lorentz gas subject to a small electrical and magnetic field. These also include Newton's law for shear viscosity for the model of two hard disks on the torus. Another major goal is the study of repellers and conditionally invariant measures for smooth chaotic maps with small holes. This aims at chaotic billiard tables with pockets and then solving the Pianigiani-Yorke conjecture of 1979. The project includes certain numerical experiments on nonequilibrium dynamics of hard disks in a box with the so-called Maxwell demon boundary conditions. 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, Maxwell in the XIX-th century. 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 ev olution of complex systems) are basic categories in the theory of chaos. Mathematical theory of chaotic attractors and repellers is still very young and quite different from the old, classical mathematics. This is one of the areas where mathematical research is currently not only most active but serves the direct needs of other modern sciences and human activities.