$\overline {d}$-continuity for countable state shifts

Abstract For full shifts on finite alphabets, Coelho and Quas [Criteria for $\overline {d}$ -continuity. Trans. Amer. Math. Soc. 350 (8) (1998), 3257–3268] showed that the map that sends a Hölder continuous potential $\phi $ to its equilibrium state $\mu _\phi $ is $\overline {d}$ -continuous. We extend this result to the setting of full shifts on countable (infinite) alphabets. As part of the proof, we show that the map that sends a strongly positive recurrent potential to its normalization is continuous for potentials on mixing countable state Markov shifts.