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.

翻译：温度混沌依赖性是指当温度趋近于某个值时吉布斯测度发散的現象。在接近绝对零度时具有混沌行为的模型存在多个基态，且这些基态均不稳定。我们研究均匀混沌模型类别，即当温度趋于零时，吉布斯测度的任意选取都会累积在整个基态集合上。我们刻画了在可计算同胚意义上均匀混沌有限范围模型可能形成的基态集合。具体而言，我们证明了每个具有有限范围和有理值相互作用的模型的基态集合在拓扑上是闭且连通的，并且属于算术层级中的 $\Pi_2$ 类。反之，每个 $\Pi_2$ 可计算的、拓扑闭且连通的概率测度集合（通过可计算同胚）均可编码为具有有限范围有理值相互作用的二维均匀混沌模型的基态集合。