  On a one-sided shift of finite type we prove that for a generic Holder
continuous function there is a unique maximizing measure. We show that b-Holder
continuous functions can be approximated in the a-Holder topology, a<b, by a
function whose maximizing measure is supported on a periodic orbit. We also
show that maximizing measures can be obtained as weak limits of equilibrium
states.
  We apply these theorems to the class of C(1+a) endomorphisms of the circle
which are coverings of degree 2, uniformly expanding and orientation
preserving. We prove that generically the invariant probability which maximizes
the Lyapunov exponent is unique and that C(1+b) endomorphisms can be
approximated in the C(1+a) topology by endomorphisms whose Lyapunov maximizing
measure is supported on a periodic orbit.
