  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 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 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.

Dynamical Systems