Non-unique topological sofic entropy and a von Neumann algebra multiplicative ergodic theorem
Yuqing Lin · 2020
A sofic approximation to a countable group is a sequence of partial actions on finite sets that asymptotically approximates the action of the group on itself by left-translations. A group is sofic if it admits a sofic approximation. Sofic entropy theory is a generalization of classical entropy theory in dynamics to actions by sofic groups. However, the sofic entropy of an action may depend on a choice of sofic approximation. All previously known examples showing this dependence rely on degenerate behavior. In joint work with D. Airey and L.Bowen an explicit example is exhibited of a mixing subshift of finite type with two different positive sofic entropies. The example is inspired by statistical physics literature on 2-coloringsof random hyper-graphs. Also, in joint work with L. Bowen and B. Hayes, the classical Multiplicative Ergodic Theorem (MET) of Oseledets is generalized to cocycles taking values in a type II von Neumann algebra. This appears to be the first MET involving operators with continuous spectrum.