The Ergodic Theory of Nonamenable Group Actions

本提案涉及不可服从群行为遍历理论中的两个主题：熵和点遍历定理。动态熵理论始于Kolmogorov（1958）的工作，他用它来研究整数群的行为。在七八十年代，熵理论首先扩展到整格的作用，然后扩展到可服从群的作用。直到2007-2008年冬季，熵理论的任何基本结果都没有被推广到任何不可服从的群体。然后，PI发现了Kolmogorov熵对群体行为的概括。这类群并不广为人知，但它包含了许多熟悉而有趣的群，包括所有可服从群和所有线性群。这被用来解决一个长期存在的开放性问题：伯努利位移的分类在一个自由群到测度共轭。在柯尔莫哥洛夫最初的工作之后，下一个重大发展是奥恩斯坦创造了一个基于熵的强大机器，用于确定整数的两个给定动作是否可测量共轭。这被扩展到可服从群体（Ornstein-Weiss），但对于不可服从群体没有类似的理论存在。本提案的一个主要研究目标是利用新的熵理论来填补这一空白。在自由群体行为的特殊情况下，与概率组合学（特别是随机规则图）和统计物理模型有密切的联系，PI打算利用这三个学科的利益。PI计划在夏威夷大学（MAanoa）举办一个研讨会，重点讨论遍历理论和动力系统中的分类问题。本次研讨会将支持本提案的智力目标，传播该领域最新的研究思想，并向年轻的研究人员和学生介绍这些主题。它也为数学界提供了一个极好的机会，鼓励大学的不同学生群体，特别是夏威夷原住民和太平洋岛民。这一建议直接为研究生的培养提供了条件。它应该特别容易吸引一个人，因为新的熵理论只需要很少的背景，并提供了大量未探索的开放问题，这些问题可能对遍历理论和其他领域产生重大影响。

BowenThis proposal is concerned with two subjects within the ergodic theory of nonamenable group actions: entropy and pointwise ergodic theorems. Dynamical entropy theory began with the work of Kolmogorov (1958) who used it to study actions of the group of integers. In the seventies and eighties, entropy theory was extended first to actions of the integer lattice, and then to actions of amenable groups. Until winter 2007-2008, none of the basic results of entropy theory had been extended to any non-amenable group. Then, the PI discovered a generalization of Kolmogorov's entropy to actions by sofic groups. This class of groups is not widely known, but it contains many familiar and interesting groups, including all amenable groups and all linear groups. This was used to solve a long-standing open problem: the classication of Bernoulli shifts over a free group up to measure-conjugacy. After Kolmogorov's initial work, the next major development was Ornstein's creation of a powerful machine, based on entropy, for determining whether two given actions of the integers are measurably-conjugate. This was extended to amenable groups (Ornstein-Weiss) but no comparable theory exists for nonamenable groups. A major research goal of this proposal is to fill this gap by taking advantage of the new entropy theory. In the special case of free group actions, there are close connections to probabilistic combinatorics (especially random regular graphs) and statistical physics models that the PI intends to exploit for the benefit of all three subjects. The PI plans to hold a workshop at the University of Hawai'i, MAanoa focussed around classication problems in ergodic theory and dynamical systems. This workshop will support the intellectual goals of this proposal, disseminate the most current research ideas in the field and introduce these topics to young researchers and students. It also provides an excellent opportunity for the mathematical community to encourage the diverse student population of the university, especially the native Hawaiian and Pacific Islanders. This proposal provides directly for the training of a graduate student. It should be particularly easy to attract one because the new entropy theory requires very little background and provides ample unexplored open problems that could have signicant impact in ergodic theory and other areas.