  We study some ergodicity property of zero-sum stochastic games with a finite
state space and possibly unbounded payoffs. We formulate this property in
operator-theoretical terms, involving the solvability of an optimality equation
for the Shapley operators (i.e., the dynamic programming operators) of a family
of perturbed games. The solvability of this equation entails the existence of
the uniform value, and its solutions yield uniform optimal stationary
strategies. We first provide an analytical characterization of this ergodicity
property, and address the generic uniqueness, up to an additive constant, of
the solutions of the optimality equation. Our analysis relies on the theory of
accretive mappings, which we apply to maps of the form $Id - T$ where $T$ is
nonexpansive. Then, we use the results of a companion work to characterize the
ergodicity of stochastic games by a geometrical condition imposed on the
transition probabilities. This condition generalizes classical notion of
ergodicity for finite Markov chains and Markov decision processes.
