  We study the zero-temperature limit of the Gibbs measures of a class of
long-range potentials on a full shift of two symbols $\{0,1\}$. These
potentials were introduced by Walters as a natural space for the transfer
operator. In our case, they are locally constant, Lipschitz continuous or, more
generally, of summable variation. We assume there exists exactly two ground
states: the fixed points $0^\infty$ and $1^\infty$. We fully characterize, in
terms of the Peierls barrier between the two ground states, the
zero-temperature phase diagram of such potentials, that is, the regions of
convergence or divergence of the Gibbs measures as the temperature goes to
zero.

Dynamical Systems