This paper studies infinite-horizon stochastic games in which players observe noisy public information about a hidden state each period. We find that if the game is connected, the limit feasible payoff set exists and is invariant to the initial prior about the state. Building on this invariance result, we provide a recursive characterization of the equilibrium payoff set and establish the folk theorem. We also show that connectedness can be replaced with an even weaker condition, called asymptotic connectedness. Asymptotic connectedness is satisfied for generic signal distributions, if the state evolution is irreducible.