  Chaotic dependence on temperature refers to the phenomenon of divergence of
Gibbs measures as the temperature approaches a certain value. Models with
chaotic behaviour near zero temperature have multiple ground states, none of
which are stable. We study the class of uniformly chaotic models, that is,
those in which, as the temperature goes to zero, every choice of Gibbs measures
accumulates on the entire set of ground states. We characterise the possible
sets of ground states of uniformly chaotic finite-range models up to computable
homeomorphisms.
  Namely, we show that the set of ground states of every model with
finite-range and rational-valued interactions is topologically closed and
connected, and belongs to the class $\Pi_2$ of the arithmetical hierarchy.
Conversely, every $\Pi_2$-computable, topologically closed and connected set of
probability measures can be encoded (via a computable homeomorphism) as the set
of ground states of a uniformly chaotic two-dimensional model with finite-range
rational-valued interactions.
