Topics in Dynamical Systems: Attractors, Dimension, Lattice Models

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The proposed research deals with problems in the theory of smooth dynamical systems and their applications to mathematical and statistical physics as well as to areas in geometry. The main subject of study is hyperbolic dynamical systems, systems exhibiting "deterministic chaos" phenomena; that is, the appearance of irregular chaotic motions in purely deterministic dynamical systems. Many global properties of a dynamical system with sufficiently strong hyperbolic behavior can be deduced from studying the linearized approximation system along its orbits. The study of this phenomena originated in seminal works of Smale, Anosov and Sinai. Such systems possess a 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 non-uniform hyperbolicity. It was introduced and studied in earlier work of the PI and is based upon the theory of separation of close orbits over time, called Lyapunov exponents. The goal of the proposed research is to develop a set of analytic tools for the subject and to construct some natural invariant measures with rich statistical properties. There are several main topics in the proposal: 1) Thermodynamic formalism for non-uniformly hyperbolic dynamical systems -- that is to establish existence and uniqueness of equilibrium measures and study phase transitions for systems with nonzero Lyapunov exponents; 2) Sinai-Ruelle-Bowen (SRB) measures for non-uniformly hyperbolic systems - that is to construct "physically natural" classes of invariant measures for dissipative systems with nonzero Lyapunov exponents; 3) Gibbs and equilibrium measures for non-uniformly hyperbolic systems constructed via a new approach that does not rely on symbolic representation of the system; 4) Essential coexistence of hyperbolic and non-hyperbolic behavior -- that is to complement the famous Kolmogorov-Arnold-Moser (KAM) theory by constructing particular examples of systems with coexistence of areas with nonzero Lyapunov exponents and areas with zero topological entropy.