Reeb vector fields and open book decompositions

Abstract

We determine parts of the contact homology of certain contact 3-manifolds in the framework of open book decompositions, due to Giroux. We study two cases: when the monodromy map of the compatible open book is periodic and when it is pseudo-Anosov. For an open book with periodic monodromy, we verify the Weinstein conjecture. In the case of an open book with pseudo-Anosov monodromy, suppose the boundary of a page of the open book is connected and the fractional Dehn twist coefficient $c$ equals $k=n$, where $n$ is the number of prongs along the boundary. If $k\ge 2$, then there is a well-defined linearized contact homology group. If $k\ge 3$, then the linearized contact homology is exponentially growing with respect to the action, and every Reeb vector field of the corresponding contact structure admits an infinite number of simple periodic orbits.