Rabinowitz Floer homology, leafwise intersections, and topological entropy

Abstract

We study dynamical properties of contact manifolds using methods from Floer theory. In the first part of this thesis we exhibit examples of contact structures on spheres of dimensions greater than $5$ having positive topological entropy. We give two different types of constructions, each requiring a different approach, each leading to positive entropy. The first approach uses the algebraic growth of wrapped Floer homology and its invariance properties under some class of contact surgeries. By carrying out a suitable series of those surgeries we then obtain contact spheres $(S^{2n-1},\xi)$ of dimensions $2n-1>5$ such that the topological entropy of every Reeb flow on $(S^{2n-1},\xi)$ is positive. Those spheres admit an exact filling by a domain that is homotopy equivalent to a bouquet of spheres. In dimension $5$ this approach leads also to the construction of a contact structure on $S^{3} \times S^{2}$ such that all its Reeb flows have positive topological entropy. The second approach uses the Floer homology of perturbations of the Rabinowitz action functional. This allows us in particular to show that there exist contact spheres in dimensions greater then $5$ that are exactly fillable by a domain diffeomorphic to a ball and such that the topological entropy of every Reeb flow on it is positive. In the second part of the thesis we define a version of Rabinowitz Floer homology for hypertight contact manifolds in symplectizations and prove versions of conjectures by Sandon and Mazzucchelli on the existence of translated points and invariant Reeb orbits. Furthermore we give a proof of the existence of non-contractible Reeb orbits on hypertight contact manifolds that admit positive loops of contactomorphisms