Mathematical Sciences: Topics in Ergodic Theory and Analysis

贝娄教授和琼斯教授将继续研究遍历理论以及更普遍的分析中出现的几乎所有收敛问题。 他们将重点关注一些问题。 其中的核心是布尔干熵准则和强扫除性质。 他们还将研究涉及薄集平均的问题，例如个体遍历定理中巴纳赫密度为零的集合、单参数流情况下的离散平均值、两参数流情况下的环等。他们计划花一些时间的一个项目涉及可积函数的几乎处处收敛。 这是一个困难的领域，但主要研究人员希望利用 Calderon-Zygmund 分解取得一些进展。 该项目涉及遍历理论的研究。 一般而言，遍历理论涉及理解系统的平均行为，这些系统的动力学过于复杂或混乱，无法在微观细节中遵循。 在“动力学”标题下可以放置关于抽象变换组如何作用于平滑空间的现代理论。 通过这种方式，遍历理论与几何学相联系，寻求对均匀空间上的流进行分类。

Professor Bellow and Jones will continue their study of almost everywhere convergence questions which arise in ergodic theory and more generally in analysis. They will focus on a number of issues. Central among them are Bourgain's entropy criterion and the strong sweeping out property. They will also study problems involving averaging over thin sets such as sets of Banach density zero in the individual ergodic theorem, discrete averages in the case of a one parameter flow, annuli in the case of a two parameter flow, etc. One project on which they plan to spend some time involves almost everywhere convergence for integrable functions. This is a difficult area but the principal investigators hope to make some progress using Calderon-Zygmund decompositions. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics" can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.