The Shannon–McMillan–Breiman theorem beyond amenable groups

Abstract

We introduce a new isomorphism-invariant notion of entropy for measure-preserving actions of arbitrary countable groups on probability spaces, which we call orbital Rokhlin entropy. It employs Danilenko’s orbital approach to entropy of a partition, and it is motivated by Seward’s recent generalization of Rokhlin’s characterization of entropy from amenable to general groups. A key ingredient in our approach is the use of an auxiliary probability-measure-preserving hyperfinite equivalence relation. Under the assumption of ergodicity of the auxiliary equivalence relation, our main result is a Shannon–McMillan–Breiman pointwise almost sure convergence theorem for the orbital entropy of partitions in measure-preserving group actions, the first such convergence result going beyond the realm of amenable groups. As a special case, we obtain a Shannon–McMillan–Breiman theorem for all strongly mixing actions of any countable group. Furthermore, we compare orbital Rokhlin entropy to Rokhlin entropy, and using an important recent result of Seward, we show that they coincide for free ergodic actions of any countable group. Finally, we consider actions of non-abelian free groups and demonstrate the geometric significance of the entropy equipartition property implied by the Shannon–McMillan–Breiman theorem. We show that the orbital entropy of a partition is the limit of the information functions of the sequence of partitions arising from refining any given finite partition along almost every horoball in the group.