r-entropy, equipartition, and Ornstein’s isomorphism theorem inR ⁿ

Abstract

A new approach is given to the entropy of a probability-preserving group action (in the context ofZ and ofR _n ), by defining an approximate “r-entropy”, 0<r<1, and lettingr → 0. If the usual entropy may be described as the growth rate of the number of essential names, then ther-entropy is the growth rate of the number of essential “groups of names” of width≦r, in an appropriate sense. The approach is especially useful for actions of continuous groups. We apply these techniques to state and prove a “second order” equipartition theorem forZ _m ×R _n and to give a “natural” proof of Ornstein’s isomorphism theorem for Bernoulli actions ofZ _m ×R _n , as well as a characterization of such actions which seems to be the appropriate generalization of “finitely determined”.