Dynamics of Partially Hyperbolic Systems

Northwestern University, Evanston IL

Investigators

Abstract

This project will investigate the dynamics of partially hyperbolic systems. It is hoped to improve the recent theorem of Pugh and Shub by weakening the center bunching hypothesis and adapting the proof so that it applies to the pointwise (or Brazilian) version of partial hyperbolicity rather than more stringent uniform assumptions made by Pugh and Shub. I also hope to extend the classes of partially hyperbolic maps within which the hypotheses of the Pugh-Shub theorem are known to hold generically by studying compact group extensions of the compact group extensions already studied by myself and Wilkinson. In addition I plan to continue my work with Paternain on magnetic flows and to collaborate with Hasselblatt and Wilkinson on a study of Lyapunov exponents for geodesic flows. This project will study the dynamics of partially hyperbolic systems. A differentiable dynamical system consists of a differentiable manifold which represents the possible states of the system and a differentiable map of the manifold to itself which represents the evolution of the system from its current state to its next state. A basic mechanism which tends to produce chaotic behavior is for the derivative of the map to stretch vectors in some directions and to shrink vectors in the complementary directions. Such behavior is called hyperbolicity. The system is called partially hyperbolic if in addition to the expanding and contracting directions that are stretched and shrunk there is a third direction which is stretched less than the expanding direction and shrunk less than the contracting. It has long been suspected that most partially hyperbolic systems should have the same chaotic behavior as fully hyperbolic systems. In the 1990's the work of Pugh and Shub (in collaboration with Grayson and Wilkinson) has made it possible to prove this in considerable generality. I aim to extend their work, by weakening the hypotheses in their main theorem and studying a number of particular examples of partially hyperbolic systems.