Canonical Reducing Curves of Surface Homeomorphism

n this paper we will introduce the concept of canonical reducing set of a surface homeomorphism, and prove that it is unique up to an isotopy. As an application, we will give a simple proof of Thurston's theorem on classifying mappings on non-orientable surface, using the techniques of quasi- conformal mappings and some known results in the orientable case, especially the Thurston theorem on orientable surfaces.