Statistical and geometric properties of dynamical systems

University Of Houston, Houston TX

Investigators

Abstract

Abstract proposal DMS-0600927 The project primarily deals with statistical and geometric properties of smooth dynamical systems, especially non-uniformly hyperbolic maps and flows. There are several interrelated topics in the proposal. A top priority is the extension of our understanding of statistical properties, such as the large deviation principle and the almost sure invariance principle for vector valued observables, to non-uniformly hyperbolic systems, in particular those modeled by a Young Tower. As part of this program, the investigators will study the regularity of group-valued measurable solutions to sub-cohomological equations and the implications of such regularity for optimal periodic orbits (periodic orbits on which Birkhoff averages of an observable are optimized). Geometric methods play an important role in the proposal. Recently, the investigators obtained results on the Whitney regularity of measurable solutions to cohomological equations on dynamically defined Cantor sets and it is proposed to extend this work to higher dimensions. The theory of equivariant transversality will be extended and applied to Hamiltonian and reversible systems, a setting important for applications. Finally, it is proposed to continue with the development and application of new methods for the analysis of mixing for hyperbolic flows. The investigators have already developed new invariants leading to a proof that every smooth hyperbolic flow can be approximated by a stably (rapid) mixing flow. It is proposed to pursue this investigation to obtain an improved understanding of exponential mixing for hyperbolic flows a topic of considerable significance in physics. Many physical systems, even those modeled accurately by deterministic differential equations, often behave in an apparently random and unpredictable way -- the phenomenon of deterministic chaos. Examples range from weather systems and fluid turbulence to electronic circuits and animal populations. Chaotic or complex systems of this type are often best understood in terms of the statistical properties of observations on the system. Complex systems may possess extra geometric structure, such as reversing symmetries or energy conservation, which can alter the expected properties of the physical system as well as provide a means for understanding the system. The proposal aims to deepen our understanding of complex systems, by investigating their statistical properties and exploiting new geometric methods to study their behavior. Results of the research will be broadly disseminated in the scientific literature and in lectures. The research will also contribute to the mathematical education of graduate students at the University of Houston, in part by lectures of the investigators, and in part by students working on this and related subjects under their guidance.