In this paper we discuss the change in contact structures as their supporting
open book decompositions have their binding components cabled. To facilitate
this and applications we define the notion of a rational open book
decomposition that generalizes the standard notion of open book decomposition
and allows one to more easily study surgeries on transverse knots. As a
corollary to our investigation we are able to show there are Stein fillable
contact structures supported by open books whose monodromies cannot be written
as a product of positive Dehn twists. We also exhibit several monoids in the
mapping class group of a surface that have contact geometric significance.Comment: 62 pages, 32 figures. Significant expansion of exposition and more
details on some argument
