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：混沌温度依赖性是指当温度趋近于特定值时，吉布斯测度出现发散的现象。具有接近零温混沌行为的模型存在多个基态，且这些基态均不稳定。我们研究了均匀混沌模型类，即当温度趋近于零时，吉布斯测度的每一个选择都收敛于整个基态集合。基于可计算同胚映射，我们刻画了均匀混沌有限范围模型可能的基态集合。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连通集，且属于算术层级中的 $\Pi_2$ 类。反之，每一个 $\Pi_2$ 可计算的拓扑闭连通概率测度集都能（通过可计算同胚映射）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。