Topics in Smooth Ergodic Theory: Stochastic Properties, Thermodynamic Formalism, Coexistence

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One of the greatest discoveries of the second half of the last century, which impacted many branches of science, was the phenomenon known as ``deterministic chaos'' – the emergence of irregular chaotic motions in purely deterministic systems. Models with this type of motions can be widely found in physics, biology, chemistry, as well as in engineering and economics. Hyperbolicity theory, which is an important part of general theory of dynamical systems, provides a mathematical foundation for the deterministic chaos phenomenon by supplying researchers with tools that allow them to describe global properties of a nonlinear dynamical system using information about infinitesimal hyperbolic behavior of its trajectories. The study of hyperbolic phenomena originated in the seminal works of Artin, Morse, Hedlund and Hopf, but the systematic study of hyperbolic dynamical systems was initiated by Smale, Anosov and Sinai, who studied dynamical systems with strong hyperbolic behavior which possess high level of unpredictability and exhibit strong chaotic behavior. In this project, the PI considers the weakest (hence, most general) form of hyperbolicity, known as non-uniform hyperbolicity. The latter originated in the work of the PI. The study of non-uniformly hyperbolic systems is based upon the theory of Lyapunov exponents, which provides some “practical” tools to detect and describe hyperbolic properties of the systems. The modern non-uniform hyperbolicity theory has numerous applications to ergodic theory, mathematical and statistical physics, Riemannian geometry, and other areas of mathematics and beyond. The project provides research training opportunities for graduate students.Building upon past results the PI will carry out a broad research program which includes the following topics: (1) Thermodynamic formalism for non-uniformly hyperbolic dynamical systems. Some ideas from geometric measure theory are used to construct equilibrium measures in hyperbolic dynamics. The new method is based on pushing forward by the dynamics the Caratheodory measure associated with the Caratheodory dimension structure generated by the potential; (2) Essential coexistence of hyperbolic and non-hyperbolic behavior. This is to understand how two different types of dynamical behavior - fully hyperbolic (positive entropy) and non-hyperbolic (zero entropy) - can coexist in an essential way. The project is aimed at constructing Hamiltonian systems and geodesic flows which exhibit the essential coexistence phenomenon thus providing new insights in the classical Kolmogorov-Arnold-Moser theory; (3) The study of two important conjectures in dynamics: (i) Katok's entropy conjecture, claiming that a volume preserving uniquely ergodic diffeomorphism has zero Kolmogorov-Sinai entropy; (ii) Baire Category conjecture, claiming that irregular sets for smooth dynamical systems and for continuous cocycles over them have the 2nd Baire Category; (4) Maps with exponential and polynomial decay of correlations on compact manifolds. The goal is to substantially advance the understanding of smooth realization problem by showing that any smooth manifold admits a volume preserving hyperbolic diffeomorphism with polynomial or exponential decay of correlations and also satisfies the Central Limit Theorem.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.