Foliated Open Books

This paper introduces a new type of open book decomposition for a contact three-manifold with a specified characteristic foliation $\mathcal{F}_\xi$ on its boundary. These \textit{foliated open books} offer a finer tool for studying contact manifolds with convex boundary than existing models, as the boundary foliation carries more data than the dividing set. In addition to establishing fundamental results about the uniqueness and existence of foliated open books, we carefully examine their relationship with the partial open books introduced by Honda-Kazez-Matic. Foliated open books have user-friendly cutting and gluing properties, and they arise naturally as submanifolds of classical open books for closed three-manifolds. We define three versions of foliated open books (embedded, Morse, and abstract), and we prove the equivalence of these models as well as a Giroux Correspondence which characterizes the foliated open books associated to a fixed triple $(M, \xi, \mathcal{F})$.