论文标题
类别理论证明了厄尔及性分解定理
A category-theoretic proof of the ergodic decomposition theorem
论文作者
论文摘要
奇异分解定理是动力学系统和千古理论的基石结果。它指出，动态系统上的每个不变措施都是千古的混合物。在这里，我们使用马尔可夫类别的形式主义，用字符串图来制定并证明定理。我们通过在随机核类别中实例化结果来恢复常规的理论陈述。一路上，我们在确定性和随机动力学系统理论中对几个概念进行了概念处理。尤其， - 奇异措施自然而然地表现为确定性形态的特定锥（从马尔可夫类别的意义上）； - 动态系统的不变$σ$ -Algebra可以看作是Markov内核类别中的colimit。 与类别理论的其他用途一致，一旦建立了必要的结构，我们的主要定理证明比传统方法要简单得多。特别是，它不使用任何定量的限制参数，也不依赖组的基数或单体索引动力学。我们希望这一结果为将类别理论进一步应用于动态系统，千古理论和信息理论的进一步应用铺平了道路。
The ergodic decomposition theorem is a cornerstone result of dynamical systems and ergodic theory. It states that every invariant measure on a dynamical system is a mixture of ergodic ones. Here we formulate and prove the theorem in terms of string diagrams, using the formalism of Markov categories. We recover the usual measure-theoretic statement by instantiating our result in the category of stochastic kernels. Along the way we give a conceptual treatment of several concepts in the theory of deterministic and stochastic dynamical systems. In particular, - ergodic measures appear very naturally as particular cones of deterministic morphisms (in the sense of Markov categories); - the invariant $σ$-algebra of a dynamical system can be seen as a colimit in the category of Markov kernels. In line with other uses of category theory, once the necessary structures are in place, our proof of the main theorem is much simpler than traditional approaches. In particular, it does not use any quantitative limiting arguments, and it does not rely on the cardinality of the group or monoid indexing the dynamics. We hope that this result paves the way for further applications of category theory to dynamical systems, ergodic theory, and information theory.
