# Zero temperature limits of Gibbs states

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Michael Hochman, Princeton University
Let $f$ be a Holder potential on the full one-sided shift ${0,1}^\mathbb{N}$, and let $\mu_b$ denote the Gibbs measure for $f$ at inverse temperature $b$ (existence and uniqueness are classical, as is the smooth dependence on $b$). It was thought that in this situation the limit of $\mu_b$ as $b$ tends to infinity should exist: although van Enter and Ruszel gave a counterexample over an infinite state space, in the finite state case Bremont proved that convergence does take place when $f$ takes on finitely many values. I will present joint work with Jean-Rene Chazottes in which we construct a counterexample and discuss some of its features. I will also discuss results in the multidimensional case.