While studying Riemann surfaces I came across this question: If $f$ :S $\rightarrow$ S is a holomorphic map between hyperbolic Riemann surfaces then $f$ at most one fixed point unless some iterate $f^{k}$ is the identity map.
Using Pick's theorem we know that if $f$ isn't a conformal automorphism or a covering map then it is strictly distance reducing and so for $f$ to have at least two fixed points it has to be either of the above two. But now I don't know how to proceed from here.
Some help with this would be highly appreciated Thanks.