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 $\sigma$-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.
翻译:egodic 脱分分解理论理论是动态系统和随机理论的基石。 它指出, 动态系统中的每一种变异性措施都是由异性体混合而成的。 在这里, 我们用 Markov 类别的形式主义来制定和证明字符串图的理论。 我们通过在随机内核类别中解析我们的结果, 恢复了通常的测量理论说明。 随着我们从概念上处理确定性和随机性动态系统理论中的若干概念的方式, 我们的主要理论证据比传统方法要简单得多。 特别是, ergodic 措施似乎非常自然地像确定性形态( 马尔科夫 类别) 的特定锥体一样; 在这里, 我们用 字符串图图来制定并证明 字符串图 ; 一个动态系统的异性 $\\ sigma$- algebra 可以被看成是Markov 内核子类别中的一种共限值。 根据类别理论的其他用途, 一旦建立了必要的结构, 我们的主要理论的证明比传统方法要简单得多。 特别是, 它不使用任何定量理论理论性理论理论理论, 我们的理论性理论性理论性理论性理论性理论性理论性理论性理论性理论性理论性理论性理论性理论性理论性理论, 它不依靠 。