Ergodic Theory for Weakly Hyperbolic Dynamical Systems

Brigham Young University, Provo UT

Investigators

Abstract

The mathematical field of dynamical systems concerns the long-term behavior of systems which evolve in time according to specified rules. Dynamical systems arise naturally in many areas of science and engineering, including statistical mechanics, neurophysiology, and climate science. This project will focus on the dynamics of certain systems – known as weakly hyperbolic systems -- that display chaotic behavior, in which small perturbations of initial conditions can lead to widely varying trajectories for the system. Because these systems are inherently difficult to predict, they are often studied from a statistical point of view, that is, one analyzes the properties of the system that are expressed through various types of average. This is the focus of ergodic theory, a subfield of dynamical systems, and the conceptual framework for this project, in which the PI will investigate the statistical properties of systems with different types of hyperbolicity. The project will also contribute to education and training, through mentorship of graduate students and the development of new seminars on the topics studied. The project has three distinct parts. Previously, the PI used measure rigidity results to identify new open sets of dynamical systems with a unique physical measure. The first part of the project aims to address questions related to the utilization of quantified non-joint integrability in establishing the existence and finiteness of physical measures, as well as to understand how often these conditions occur in the partially hyperbolic setting. In the second part of the project, the PI and his co-authors aim to understand different types of transversality to obtain absolute continuity of stationary measures for certain types of random products of surface diffeomorphisms. One goal of this part is to obtain a Benoist-Quint type of result in a non-homogeneous setting. The third part of the project focuses on applying coding techniques to study measures of maximal entropy for non-invertible systems possibly having singularities. Some of the goals include understanding conditions that guarantee existence and finiteness of measures of maximal entropy in these settings, and understanding new examples of such maps. 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.