Partial Hyperbolicity and Rigidity

Northwestern University, Evanston IL

Investigators

Abstract

In the past few decades, the topic of partially hyperbolic dynamical systems has emerged as one main direction in which the theory of complicated ("chaotic") dynamical systems has extended beyond the classical setting of hyperbolic dynamics. This research has been driven in part by the realization that many practical applications of dynamical systems to experimental phenomena require a much broader theory. The PI's research over the last decade has focused on the fundamental ergodic -- that is, statistical -- properties that partially hyperbolic systems generically can be expected to display. The PI's work with her collaborators has led recently to a proof of a "Boltzmann Ergodic Hypothesis" for partially hyperbolic systems in low dimension: the generic conservative partially hyperbolic system is ergodic. The proposed research has two aspects: * Extending the previous work of the PI and her collaborators on the ergodicity of partially hyperbolic diffeomorphisms. * A study of rigidity of solvable groups actions on manifolds, in particular those actions that contain partially hyperbolic elements. The material resulting from this research proposal will be widely disseminated, through talks at conferences, online preprint servers, and publication in scholarly journals. A significant component of this grant is devoted to graduate-student training on the Ph.D. level.