  We study the equilibrium behaviour of a two-sided topological Markov shift
with a countable number of states. We assume the potential associated with this
shift is Walters with finite first variation and that the shift is
topologically transitive. We show the equilibrium measure of the system is
Bernoulli up to a period. In the process we generalize several theorems on
countable Markov shifts. We prove a variational principle and the uniqueness of
equilibrium measures. A key step is to show that functions with Walters
property on a two-sided shift are cohomologous to one-sided functions with the
Walters property. Then we turn to show that functions with summable variations
on two-sided CMS are cohomologous to one-sided functions, also with summable
variations.
