A Wolff-Denjoy theorem for infinitely connected Riemann surfaces

We generalize the classical Wolff-Denjoy theorem to certain infinitely connected Riemann surfaces. Let $X$ be a non-parabolic Riemann surface with Martin boundary $\Delta$. Suppose each Martin function $k_{y}$, $y\in \Delta$, extends continuously to $\Delta \setminus \{y\}$ and vanishes there. We show that if $f$ is an endomorphism of $X$ and the iterates of $f$ converge to the point at infinity, then the iterates converge locally uniformly to a point in $\Delta$. As an application, we extend the Wolff-Denjoy theorem to non-elementary Gromov hyperbolic covering spaces of compact Riemann surfaces. Such covering surfaces are of independent interest. Finally, we use the theory of non-tangential boundary limits to give a version of the Wolff-Denjoy theorem that imposes certain mild restrictions on $f$ but none on $X$ itself.