  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 $c={k\over n}$, where $n$ is the number of prongs along
the boundary. If $k\geq 2$, then there is a well-defined linearized contact
homology group. If $k\geq 3$, 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.
