Model Theory and Ergodic Theorems

遍历理论是数学的一个分支，研究的是动力系统在长时间间隔内运行时的行为。遍历理论的基本结果之一是冯·诺依曼的平均遍历定理（1932），它相当于关于保守系统的极限平均状态的抽象陈述。在这个项目中，我们寻求应用模型论的概念和技术，数学逻辑的一个分支，通过重新解释和扩展冯诺依曼之后几十年的研究结果，建立遍历平均和遍历递归的结果。模型理论的观点应该有助于阐明和潜在地揭示遍历理论和其他数学分支之间的联系。该项目的目标之一是将最近关于多个遍历平均值收敛的结果转换为一个合适的模型理论框架，该框架允许将分析参数与类型理论、分叉演算和序数秩相结合。一个重要的概念，在拟议的研究是排名的复杂性“多项式行动”的一组G的一些措施空间X，或更确切地说，诱导（多项式）行动的G各种拓扑空间（说，X上的功能）。这就需要开发一个抽象的代数框架，扩展Leibman的理论之间的多项式映射组。辅助预期产品的模型理论方法包括亚稳态收敛的结果（类似于一致收敛不存在的情况下的弱形式的一致性），关于多项式遍历系统的超积的平均值的收敛，以及关于序数的多项式遍历作用。（超限）秩（广义Leibman度）。连接和应用组合，拉姆齐理论，数论也将研究。

Ergodic theory is a branch of mathematics concerned with the behavior of dynamical systems when they are allowed to run over long time intervals. One of the fundamental results of ergodic theory is the Mean Ergodic Theorem of von Neumann (1932), which amounts to an abstract statement about the limiting average state of a conservative system. In this project we seek to apply concepts and techniques of model theory, a branch of mathematical logic, to establish results on ergodic averages and ergodic recurrence by reinterpreting and extending research results discovered in the decades following von Neumann. The model-theoretic viewpoint should help illuminate and potentially unveil connections between ergodic theory and other branches of mathematics.One of the project's goals is casting recent results on convergence of multiple ergodic averages into a suitable model-theoretic framework that allows combining analytic arguments with the theory of types, forking calculus, and ordinal ranks. An important concept in the proposed research is a ranking of the complexity of "polynomial actions" of a group G on some measure space X, or rather of the induced (polynomial) actions of G on various topological spaces (say, of functions on X). This requires developing an abstract algebraic framework that extends Leibman's theory of polynomial mappings between groups. Ancillary anticipated products of the model-theoretic approach include results on metastable convergence (akin to a weak form of uniformity in situations in which uniform convergence is absent), on convergence of averages on ultraproducts of polynomial-ergodic systems, and on polynomial-ergodic actions of ordinal (transfinite) rank (generalized Leibman degree).Connections and applications to combinatorics, Ramsey theory, and number theory will also be studied.