The Ergodic Theory of Nonamenable Group Actions

鲍文这个建议涉及不可服从群作用的遍历理论中的两个主题：熵和逐点遍历定理。动力熵理论始于科尔莫戈洛夫(1958)的工作，他用它来研究整数群的作用。在七八十年代，熵理论首先被扩展到整数格的作用，然后扩展到顺从群的作用。在2007-2008年冬季之前，没有一个熵理论的基本结果被推广到任何不可服从的群体。然后，PI发现了Kolmogorov熵对SOFIC群作用的推广。这类群并不广为人知，但它包含了许多熟悉而有趣的群，包括所有从属群和所有线性群。这被用来解决一个长期悬而未决的问题：伯努利的经典在自由群上转移到测度共轭。在科尔莫戈罗夫最初的工作之后，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共轭。这被扩展到顺从群(Ornstein-Weiss)，但不存在适用于非顺从群的类似理论。这一提议的一个主要研究目标是利用新的熵理论来填补这一空白。在自由群作用的特殊情况下，与概率组合学(特别是随机正则图)和统计物理模型有密切的联系，PI打算利用这些模型来使所有三个对象受益。国际和平协会计划在马阿诺阿的夏威夷大学举办一次研讨会，重点讨论遍历理论和动力系统中的经典问题。这个研讨会将支持这一提议的智力目标，传播该领域最新的研究想法，并向年轻的研究人员和学生介绍这些主题。它还为数学界提供了一个极好的机会，鼓励该大学多样化的学生群体，特别是夏威夷土著和太平洋岛民。这项建议直接规定了研究生的培训。这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，这些问题可能会在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.