rentropy, equipartition, and Ornstein’s isomorphism theorem inR ^{ n }
 Jacob Feldman
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A new approach is given to the entropy of a probabilitypreserving group action (in the context ofZ and ofR _{ n }), by defining an approximate “rentropy”, 0<r<1, and lettingr → 0. If the usual entropy may be described as the growth rate of the number of essential names, then therentropy 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”.
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 Title
 rentropy, equipartition, and Ornstein’s isomorphism theorem inR ^{ n }
 Journal

Israel Journal of Mathematics
Volume 36, Issue 34 , pp 321345
 Cover Date
 19800901
 DOI
 10.1007/BF02762054
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 SpringerVerlag
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 Authors

 Jacob Feldman ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of California, 94720, Berkeley, Calif., USA