Cohomology, Skew-Products, and Partially Hyperbolic Diffeomorphisms

West Chester University Of Pennsylvania, West Chester PA

Investigators

Abstract

The proposed research has several goals. The first goal is to study cohomological equations over hyperbolic dynamical systems. One is interested to generalize Livsic's cohomological results to cocycles for which periodic data is included in semigroups in Lie groups. Positive results in this direction will be natural generalizations of previous work, and will give obstructions to the topological transitivity of extensions over hyperbolic actions. The second goal is to use cohomological results as a tool in the rigidity theory of partially hyperbolic higher rank lattice actions. This goal is part of the rigidity program initiated by Zimmer, aiming to classify volume preserving higher rank lattice actions on compact manifolds. A reachable target is to classify small GL(n,R) extensions of SL(n,Z) totally non-symplectic actions (i.e. hyperbolic and with a special structure of the stable and unstable foliations). The third goal is to find generic classes of partially hyperbolic transformations with rich dynamic properties. Pugh and Shub recently conjectured that stably ergodicity and accessibility occur more frequently than expected among differentiable maps, and that some partial hyperbolicity is sufficient to prove stably ergodicity. From a result of Nitica and Torok it follows that, under technical conditions, stably ergodicity is open and dense in the class of partially hyperbolic diffeomorphisms with one-dimensional central foliation. The density holds even in the higher regularity classes. An interesting problem is to generalize this result to more general classes of partially hyperbolic diffeomorphisms. In other direction, a recent result of Nitica and Pollicott shows that for Euclidean extensions of Anosov diffeomorphisms on infranilmanifolds the only obstructions to stable transitivity are of cohomological nature. One would like to generalize this result to non-abelian fibers, and to classify the obstructions to topological transitivity. The Investigator (with I. Melbourne and A. Torok) has recent results in this direction proving the existence of stably transitive extensions with Sp(n) fiber. A chaotic map has a dense set of periodic points, that is points that are fixed by a higher iterate of the map, as well as transitive points, that is points for which a higher iterate will get arbitrarily close to any other point. Chaotic behaviour is expected to be generic in large classes of maps and in nature. Part of this research will be focused on finding new mechanisms for producing chaos for large classes of dynamical systems that exhibit only partial hyperbolicity. It will also concentrate on finding obstructions to chaotic behaviour. These results will be of interest to the broader scientific community involved in applications of nonlinear dynamics in physical sciences. A spin-off of this work will be a careful study of the semigroups in the Special Euclidean groups with interesting applications to discrete control theory and robotics. The Investigator actively participates in recruitment, training and professional development of K-12 mathematics teachers. He will continue his collaboration with undergraduate students and will support them to give talks at professional meetings. These activities will benefit from the grant.