Hyperbolic behavior in dynamical systems: Higher-order phenomena beyond uniformity

University Of Houston, Houston TX

Investigators

Abstract

The mathematical theory of chaotic systems describes how randomness arises from "sensitive dependence on initial conditions": a small error in measuring the current state of a system can translate into a large error in a prediction of the system's future. Systems that behave this way can be studied using the mathematical tools of probability theory. These probabilistic descriptions depend on what is called an "invariant measure", which represents the likelihood of observing different types of behavior. For example, if a die is rolled repeatedly, one invariant measure might say that all six numbers are equally likely, while another invariant measure might say that in the long run, twice as many sixes will appear as ones. If the die is fair, then the first invariant measure is the one to trust, but if it is unevenly weighted, the second could apply. One goal of the present project is to gain a better understanding of the set of all possible invariant measures for a given system. This is an important part of making valid predictions for systems with chaotic behavior. Part of this project also involves the training of graduate students and the development of a graduate textbook on "Ergodic Theory and Hyperbolic Dynamics". In more technical terms, dynamical systems with hyperbolic ("chaotic") behavior can be studied as stochastic processes by equipping them with invariant measures and using the tools of ergodic theory. There are generally many invariant measures; thermodynamic formalism provides tools for identifying the measures that most fully capture the complexity of the system. A wide range of phenomena have been observed in the ergodic theory of hyperbolic systems: some (first-order) primarily concern a single invariant measure, while others (higher-order) involve multiple invariant measures in an essential way. Both types of phenomena have been well-studied for uniformly hyperbolic systems, but beyond uniform hyperbolicity the picture is far from complete. The present project centers on the study of higher-order phenomena beyond uniform hyperbolicity, extending current techniques and developing new ones. The condition of uniform hyperbolicity is restrictive, and many interesting systems encountered "in the wild" involve non-uniform hyperbolicity, the presence of singularities, or both. This includes fundamental examples such as dispersing billiards and Lorenz flow, as well as geometrically significant examples such as geodesic flow on manifolds beyond negative curvature. The PI has developed powerful methods for studying thermodynamic formalism of non-uniformly hyperbolic systems, and plans to extend the reach of these methods to give a more complete picture of hyperbolic behavior for a wide range of important systems. Higher-order phenomena are found in the geometry of the space of equilibrium measures; effective uniqueness results in thermodynamic formalism; connections between thermodynamic formalism and moduli space of hyperbolic systems; ubiquity of adapted and non-adapted measures; and the coexistence of hyperbolic and nonhyperbolic behavior. Progress on such questions would substantially advance our understanding of the mechanisms driving hyperbolicity and the appearance of stochastic behavior in deterministic dynamics. 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.