Topics in Dynamical Systems: Attractors, Dimension, Lattice Model

Abstract. The project outlines scientific activity in three directions. The first is the theory of coupled map lattices (CML) where research is focused on the thermodynamical limit for infinite-dimensional SRB measures (in the sense of Sinai); stability of traveling wave solutions, and transition from CMLs to PDEs via traveling waves. It also includes the study of some well-known PDEs (for example, Swift-Hohenberg equation which is of great interest in neurobiology). The second direction is concerned with the dimension theory of dynamical systems and includes the multifractal analysis of Birkhoff averages with applications to some problems in number theory as well as the multifractal rigidity phenomenon for conformal expanding maps. The third direction of research deals with some recent advantages in smooth ergodic theory. In particular, it is proposed to construct some new examples of volume preserving diffeomorphisms with non-zero Lyapunov exponents (on any Riemannian manifold and with countable number of ergodic components). It is also proposed to effect a multifractal formalism for two-dimensional hyperbolic measures. The main goal of the proposed research is to develop further the mathematical theory of chaos. It deals with various aspects of this theory including the appearance of chaotic motions in dynamical systems (which are mathematical models of various phenomena in physics, biology, economics, etc. ) and the interplay between chaotic regimes and fractal geometric structure of the space. It is proposed to study some macro-characteristics of chaotic behavior (such as entropy and Lyapunov exponents) and relate them to fractal dimension of the space. It is also proposed to construct some new examples of chaotic systems both conservative and dissipative, finite-dimensional as well as infinite-dimensional. These examples may serve as models of such extremely complicated phenomena as turbulence in hydrodynamics, neuron and memory activity in neuroscience, plant growth in plant biology, etc.