CAREER: Ergodic Theory of Nonamenable Group Actions

这个项目涉及三个主题内的遍历理论的不服从群体行动：熵，刚性/柔性现象，和逐点遍历定理。 自柯尔莫哥洛夫的首次突破（1958年）以来，动力学熵在顺从行为的遍历理论中发挥了重要作用。校长最近发现了柯尔莫哥洛夫熵的一个推广，以sofic集团的行动。这类群包含许多熟悉和有趣的群，包括所有的线性群，从而提供了熵理论的第一个扩展到某些不服从群。这一新的发展为大量可能在遍历理论和其他领域产生重大影响的问题打开了大门（例如，统计物理学、概率组合学、算子代数）。在过去的十年里，人们对不服从的群体行为的刚性理论产生了极大的兴趣（例如，见A.Furman，D. Gaboriau和S. Popa）。新的结果表明，在特殊条件下，测度共轭等价于作用量的先验较弱等价形式，如轨道等价和von Neumann等价。预计首席研究员的新熵理论可以作出重要贡献的刚性理论更精确地确定当它是一个轨道等价或冯诺依曼等价不变。首席研究员最近证明了第一个轨道等效灵活性的结果，在不受约束的设置。预计为证明而开发的技术将得到更广泛的应用，并可能导致对某些群体和行动的一般轨道等效灵活性理论。点态遍历定理有很长的历史，从伯克霍夫开始。与A合作。内沃，主要研究员是发展一般方法证明逐点遍历定理的行动离散，nonaffordable组。到目前为止的成功包括新的逐点遍历平均序列的行动的自由群体和第一个逐点遍历定理的行动一般字双曲群就球和球面平均数。新的技术消除了一个根本的障碍，因为它们减少了问题的顺从的情况下，然后可以解决经典的维纳-卡尔德龙-福尔纳理论。该项目的教育部分包括每年在得克萨斯州的研讨会，目的是向研究生介绍最新的研究进展，并汇集在该项目重要领域工作的不同研究人员群体。这位首席研究员计划通过他所在系的荣誉项目指导本科生，并在数学俱乐部会议上以及在他所在系的本科生和高中生暑期项目中发表受欢迎的演讲。首席研究员还计划邀请一位专家评审作为杰出讲师，就当前感兴趣的主题进行一系列鼓舞人心的演讲。该项目直接提供研究生的培训。它应该很容易吸引一个，因为主要研究者的最新成果是可访问的，并导致一个开放的领域的问题，其解决方案可能会有重大影响，在遍历理论和其他领域。

This project is concerned with three subjects within the ergodic theory of nonamenable group actions: entropy, rigidity/flexibility phenomena, and pointwise ergodic theorems. Since Kolmogorov's initial breakthrough (1958), dynamical entropy has played a major role in the ergodic theory of amenable actions. The principal recently discovered a generalization of Kolmogorov's entropy to actions by sofic groups. This class of groups contains many familiar and interesting groups, including all linear groups, thereby providing the first extension of entropy theory to certain nonamenable groups. This new development opens the door to an abundance of problems that could have a great impact in ergodic theory and other fields (e.g., statistical physics, probabilistic combinatorics, operator algebras). In the last decade, there has been an explosion of interest in rigidity theory for nonamenable group actions (see, for example, works of A.Furman, D. Gaboriau, and S. Popa). New results indicate that, under special conditions, measure-conjugacy is equivalent to a priori weaker forms of equivalence of actions, such as orbit equivalence and von Neumann equivalence. It is expected that the principal investigator's new entropy theory can make an important contribution to rigidity theory by determining more precisely when it is an orbit equivalence or von Neumann equivalence invariant. The principal investigator has recently proven the first orbit equivalence flexibility results in the nonamenable setting. It is expected that the techniques developed for the proof will be more broadly applicable and could result in a general orbit equivalence flexibility theory for certain groups and actions. Pointwise ergodic theorems have a long history, beginning with Birkhoff. In collaboration with A. Nevo, the principal investigator is developing general methods for proving pointwise ergodic theorems for actions of discrete, nonamenable groups. Successes so far include new pointwise ergodic averaging sequences for actions of free groups and the first pointwise ergodic theorems for actions of general word-hyperbolic groups with respect to ball and spherical averages. The new techniques remove a fundamental obstacle in that they reduce the problem to the amenable case, which can then be solved by the classical Wiener-Calderon-Folner theory. The educational component of the project includes yearly workshops at Texas A&M for the purposes of introducing graduate students to the latest research developments and bringing together the diverse groups of researchers working in areas important to the project. The principal investigator plans to mentor undergraduates through his department's honors program and to give popular talks at math club meetings and during his department's summer programs for undergraduates and high school students. The principal investigator also plans to invite an expert expositor as a distinguished lecturer to give a series of inspiring talks on topics of current interest. This project directly provides for the training of a graduate student. It should be easy to attract one because the principal investigator's recent results are accessible and lead to an open field of problems whose solutions could have significant impact in ergodic theory and other areas.