  One of the main problems of the theory of dynamical systems is the
determination of the existence of periodic orbits of a self-map and more
generally, the structure of the set of periods. Define the minimum period of a
class os self-maps of a fixed set as the minimum of the positive integers such
that each map in the class has a periodic point whose period is at most this
number.
  The problem of the determination of the minimum period of the classes of
homeomorphisms of closed surfaces was completely solved, in successive steps,
from 1910 to 1996. The aim of our work is, for each compact, connected,
orientable surface, determine the minimum period of its class of
homeomorphisms. If the genus of the considered surface is zero or on, then the
problem can be solved by simple techniques. For the case of genus at least two,
we have found two upper bounds for the minimum periods, which can be expressed
as a linear function of the genus and the number of boundary components of the
surface. We give certain sufficient conditions under which these upper bounds
are achieved. In particular, we have proved that the minimum period becomes
constant for each genus, provided that the number of boundary components is
large enough. We have also studied the minimum periods of the classes of
finite-order maps.
  This thesis has three branches which are interconnected. One has to do with
the application of the fixed-point theory. One of the upper bounds of the
minimum periods is a consequence of this theory. To obtain the other upper
bound, we have also applied the Thurston-Nielsen classification of
homeomorphisms of surfaces and some of its consequences. This is the second
branch. Finally, the third branch has to do with the theory of planar
discontinuous group which provide us with the necessary tools for the
construction of examples which prove the existence of lower bounds of the
minimum periods.
