This 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.
