  This thesis provides a classification of generalized pseudo-Anosov
homeomorphisms up to topological conjugacy using an algorithmic approach. A
Markov partition of a generalized pseudo-Anosov homeomorphism is a
decomposition of the surface into a finite number of rectangles with disjoint
interiors, such that their images intersect with any other rectangle in the
Markov partition along a finite number of horizontal sub-rectangles. Every
generalized pseudo-Anosov homeomorphism has a Markov partition, and, by using
the surface's orientation, we can endow any Markov partition with a
geometrization. The geometric type of a geometric Markov partition was defined
by Bonatti and Langevin in their book, "Diffeomorphismes de Smale des
surfaces", to classify saddle-type basic pieces for structurally stable
diffeomorphisms on surfaces. A geometric type is an abstract combinatorial
object that generalizes the incidence matrix of a Markov partition. It takes
into account not only the number of times the image of a rectangle intersects
with any other rectangle in the family but also the order and change of
orientation induced by the homeomorphisms. This thesis employs the geometric
type of a geometric Markov partition to classify conjugacy classes of
pseudo-Anosov homeomorphisms. The classification is provided by the three main
results in this manuscript: I) The geometric type is a complete invariant of
conjugation. II) A criterion is provided for determining whether an abstract
geometric type is realized by a geometric Markov partition of a pseudo-Anosov
homeomorphism. III) An algorithm is described for determining whether two
geometric types in the pseudo-Anosov class are realized by generalized
pseudo-Anosov homeomorphisms that are topologically conjugated or not.
