  We study the ergodicity of deterministic two-person zero-sum differential
games. This property is defined by the uniform convergence to a constant of
either the infinite-horizon discounted value as the discount factor tends to
zero, or equivalently, the averaged finite-horizon value as the time goes to
infinity. We provide necessary and sufficient conditions for the unique
ergodicity of a game. This notion extends the classical one for dynamical
systems, namely when ergodicity holds with any (suitable) perturbation of the
running payoff function. Our main condition is symmetric between the two
players and involve dominions, i.e., subsets of states that one player can make
approximately invariant.
