Topics in Dynamical Systems: Attractors, Dimension, Lattice Model

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The proposed research deals with problems in Smooth Dynamical Systems and their applications to Ergodic Theory, Mathematical and Statistical Physics and Geometry. The main subject of study is dynamical systems with sufficiently strong instability of trajectories -- the so-called hyperbolic dynamical systems. Hyperbolic theory provides 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 systems along its trajectories. The study of hyperbolic phenomena originated in seminal works of Artin, Morse, Hedlund, and Hopf on the instability and ergodic properties of geodesic flows on compact surfaces. Later, hyperbolic behavior was observed in other situations (for example, Smale horseshoes and hyperbolic toral automorphism). The systematic study of hyperbolic dynamical systems was initiated by Smale, Anosov and Sinai who studied dynamical 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. There are three main topics in the proposal. 1. Thermodynamics of nonuniformly hyperbolic systems -- this is to build statistical physics of phase transitions for systems with nonzero Lyapunov exponents based on recent works on Markov extensions and tower constructions. 2. Dynamical systems with mixed hyperbolicity and stable ergodicity -- this is to study how "typical" the systems with nonuniform hyperbolic behavior are. A recent result by Dolgopyat and the PI shows that such systems exist on any phase space. 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. The PI also proposes to apply his work to the FitzHugh-Nagumo equation -- the famous model that describes the propagation of voltage impulse through a nerve axon. This provides interesting new and "naturally" appearing examples of nonuniformly hyperbolic "strange" attractors as well as demonstrates transition from relatively simple Morse-Smale systems to "strange" attractors to Smale horseshoes.