The ShannonMcMillanBreiman theorem beyond amenable groups
Abstract
We introduce a new isomorphisminvariant 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 is motivated by Seward's recent generalization of Rokhlin entropy from amenable to general groups. A key ingredient in our approach is the use of an auxiliary probabilitymeasurepreserving hyperfinite equivalence relation. Under the assumption of ergodicity of the auxiliary equivalence relation, our main result is a ShannonMcMillanBreiman pointwise almost sure convergence theorem for the orbital entropy of partitions in measurepreserving group actions, the first such convergence result going beyond the realm of amenable groups. As a special case, we obtain a ShannonMcMillanBreiman 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 nonAbelian free groups and demonstrate the geometric significance of the entropy equipartition property implied by the ShannonMcMillanBreiman 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 arXiv:
 arXiv:1701.03965
 Bibcode:
 2017arXiv170103965N
 Keywords:

 Mathematics  Dynamical Systems
 EPrint:
 Additional references and attributions, changes to the introduction, and changes of terminology. No changes to the main results or their proofs