Each of the resulting compact surfaces with boundary is acted upon by some power (i.e. iterated composition) of g, and the classification can again be applied to this homeomorphism.
In fact, the proof of the classification theorem leads to a canonical representative of each mapping class with good geometric properties.
The mapping torus Mg of a homeomorphism g of a surface S is the 3-manifold obtained from S × [0,1] by gluing S × {0} to S × {1} using g. If S has genus at least two, the geometric structure of Mg is related to the type of g in the classification as follows: The first two cases are comparatively easy, while the existence of a hyperbolic structure on the mapping torus of a pseudo-Anosov homeomorphism is a deep and difficult theorem (also due to Thurston).
Fibered hyperbolic 3-manifolds have a number of interesting and pathological properties; for example, Cannon and Thurston showed that the surface subgroup of the arising Kleinian group has limit set which is a sphere-filling curve.
The three types of surface homeomorphisms are also related to the dynamics of the mapping class group Mod(S) on the Teichmüller space T(S).