Partially Hyperbolic Diffeomorphisms, Foliations, and Flows

The field of dynamical systems studies properties of functions under repeated iterations, including their periodic and long-time behavior. Dynamical systems have many applications, for example to predict weather behavior, such as hurricanes. These systems follow certain laws and the dynamics predict future behavior given an initial set of conditions.

The principal investigator will study a certain class of dynamical systems where the ambient space is locally three dimensional. In particular, the project will study what are called partially hyperbolic diffeomorphisms. The importance of these systems is that they satisfy strong stability conditions: nearby systems are also partially hyperbolic, and they are the only ones in dimension three satisfying these properties.

The project will analyze the structure of partially hyperbolic diffeomorphisms in dimension three, an area that is currently extremely active. The project will also involve the training of graduate students.

Partially hyperbolic diffeomorphisms (in dimension three) are bijective smooth maps that admit invariant contracting and expanding directions, and another invariant direction that is in between, called the center direction. One goal of the project is to understand these diffeomorphisms when they are homotopic to the identity. In particular, this includes all such maps in hyperbolic 3-manifolds, up to iterates.

Hyperbolic 3-manifolds are by far the most common manifolds in dimension three. The project will also analyze the new class of collapsed Anosov flows, recently introduced by the PI together with T. Barthelme and R.

Potrie. This class includes all known transitive examples of partially hyperbolic diffeomorphisms in dimension three, when the fundamental group is not virtually solvable. One goal is to prove that in certain classes of manifolds, including all Seifert manifolds with hyperbolic base, the family of collapsed Anosov flows includes all partially hyperbolic diffeomorphisms.

The project will also analyze the fundamental property of ergodicity for such diffeomorphisms. The PI recently showed that a large class of such diffeomorphisms is ergodic. The project aims to show ergodicity for all partially hyperbolic diffeomorphisms in certain manifolds, including all Seifert manifolds with hyperbolic base.

The project also aims to show ergodicity for all collapsed Anosov flows not equivalent to suspensions.

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.