  Pointwise convergence of spherical averages is proved for a
measure-preserving action of a Fuchsian group. The proof is based on a new
variant of the Bowen-Series symbolic coding for Fuchsian groups that,
developing a method introduced by Wroten, simultaneously encodes all possible
shortest paths representing a given group element. The resulting coding is
self-inverse, giving a reversible Markov chain to which methods previously
introduced by the first author for the case of free groups may be applied.
