Topics in Dynamical Systems: Attractors, Dimension, Lattice Models

There are several main topics in the proposal: 1) Thermodynamic formalism for nonuniformly hyperbolic dynamical systems -- this is to build statistical physics of phase transitions for systems with nonzero Lyapunov. 2) SRB measures for nonuniformly hyperbolic systems - this is to to build 'physically natural' class of invariant measures for dissipative systems. 3) Coexistence of hyperbolic and non-hyperbolic behavior -- this is to complement the famous Kolmogorov-Arnold-Moser (KAM) theory by constructing particular examples of systems with coexistence of nonzero Lyapunov exponents and areas with zero entropy. 4) Anosov rigidity - this is to establish a subtle relation between uniform and nonuniform types of hyperbolicity. 5) Dimension of non-conformal repellers - this is to study Hausdorff dimension for generic non-conformal repellers.

The proposed research deals with problems in the theory of smooth dynamical systems and their applications to mathematical and statistical physics and geometry. The main subject of study is hyperbolic dynamical systems that provide a mathematical foundation for the paradigm that is widely known as ``deterministic chaos'' -- the appearance of irregular chaotic motions in purely deterministic dynamical systems. This paradigm asserts that conclusions about global properties of a nonlinear dynamical system with sufficiently strong hyperbolic behavior can be deduced from studying the linearized system along its trajectories. The study of hyperbolic phenomena originated in seminal works of Artin, Morse, Hedlund, and Hopf on ergodic properties of geodesic flows on compact surfaces. Later, hyperbolic behavior was observed in other situations (e,g, Smale horseshoes and hyperbolic toral automorphism). The systematic study of hyperbolicity was initiated by Smale, Anosov and Sinai who studied systems with sufficiently strong hyperbolic behavior. Such systems possess high level of unpredictability and exhibit strong chaotic behavior. In the proposal the PI considers the weakest (hence, most general) form of hyperbolicity known as nonuniform hyperbolicity. The theory of nonuniformly hyperbolic dynamical systems originated in the work of the PI (sometimes this theory is referred to as ``Pesin theory'') and the study of these systems is based upon the theory of Lyapunov exponents. The PI's theory has a wide impact on the entire field. The PI will work with graduate students during the period of this award.