An entropy-preserving Dye’s theorem for ergodic actions

This work shows that equality of entropy for ergodic actions of a discrete amenable group is a restricted orbit equivalence in the formal sense defined inRestricted Orbit Equivalence for Actions of Discrete Amenable Groups by Kammeyer and Rudolph [3]. An element of the full-group of such an action encodes a countable partition. A natural extension of entropy to such countable partitions is shown to be a size in the sense of [3] and hence engenders and orbit equivalence relation on the space of all such actions. The major goal achieved here is to show that this relation is precisely equality of entropy.