Symplectic and contact aspects of foliations and Anosov flows in dimension three

Abstract

This dissertation studies the interactions between foliations and contact structures and the symplectic geometry of Anosov flows in dimension three. It is divided into two parts.In the first part, we detail a new construction of codimension-one foliations on three-manifolds from suitable pairs of contact structures. This constitutes a converse result to a celebrated theorem of Eliashberg and Thurston about approximations of foliations by contact structures. Whereas foliations are rather rigid objects, this contact viewpoint reveals some surprising flexibility and provides new insight on the L-space conjecture. The second part focuses on an important class of foliations arising from Anosov flows. We first show that three-dimensional Anosov flows can be entirely characterized in terms of symplectic geometry. As a result, we obtain new invariants for Anosov flows coming for Floer theory. We then explore the structure of these invariants and compute them explicitly in relevant cases. This is based on joint work with Kai Cieliebak, Oleg Lazarev, and Agustin Moreno, and it constitutes the first thorough investigation of Liouville manifolds beyond the Weinstein case. We finally sketch a proof that orbit equivalent Anosov flows define equivalent Liouville structures, implying that our Floer-theoretic invariants are topological invariants of Anosov flows.