On the zero-temperature limit of gibbs states

We exhibit Lipschitz (and hence Hölder) potentials on the full shift {0,1}^N such that the associated Gibbs measures fail to converge as the temperature goes to zero. Thus there are "exponentially decaying" interactions on the configuration space {0,1}^Z for which the zero-temperature limit of the associated Gibbs measures does not exist. In higher dimension, namely on the configuration space {0,1}^Zd, d ≥ 3, we show that this non-convergence behavior can occur for the equilibrium states of finite-range interactions, that is, for locally constant potentials.