Reparametrization of -flows of zero entropy

Authors:
J. Feldman and D. Nadler

Journal:
Trans. Amer. Math. Soc. **256** (1979), 289-304

MSC:
Primary 28D15

DOI:
https://doi.org/10.1090/S0002-9947-1979-0546919-6

Correction:
Trans. Amer. Math. Soc. **264** (1981), 583-585.

MathSciNet review:
546919

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Abstract | References | Similar Articles | Additional Information

Abstract: Let , be two ergodic *n*-parameter flows which preserve finite probability measures on their spaces *X*, *Y*. Let *T* be a nullset-preserving map: sending each -orbit homeomorphically to a -orbit. Then , are called homeomorphically orbit-equivalent. For , there has been developed a theory of such equivalence: ``Loosely Bernoulli'' theory. A completely parallel theory exists for higher dimensions, except that it is necessary to impose a certain natural ``growth'' restriction on *T*, a restriction which is vacuous in the case . In this paper we carry out this program, but only for the case of zero entropy.

**[A]**L. M. Abramov,*The entropy of a derived automorphism*, Dokl. Akad. Nauk SSSR**128**(1959), 647–650 (Russian). MR**0113984****[D]**H. A. Dye,*On groups of measure preserving transformation. I*, Amer. J. Math.**81**(1959), 119–159. MR**0131516**, https://doi.org/10.2307/2372852**[F1]**J. Feldman,*New 𝐾-automorphisms and a problem of Kakutani*, Israel J. Math.**24**(1976), no. 1, 16–38. MR**0409763**, https://doi.org/10.1007/BF02761426**[F2]**-,*r-entropy, equipartition, and Ornstein's isomorphism theorem in*(to appear).**[K]**Shizuo Kakutani,*Induced measure preserving transformations*, Proc. Imp. Acad. Tokyo**19**(1943), 635–641. MR**0014222****[Ka]**A. B. Katok,*Time change, monotone equivalence, and standard dynamical systems*, Dokl. Akad. Nauk SSSR**223**(1975), no. 4, 789–792 (Russian). MR**0412383****[L]**D. A. Lind,*Locally compact measure preserving flows*, Advances in Math.**15**(1975), 175–193. MR**0382595**, https://doi.org/10.1016/0001-8708(75)90133-4**[N]**D. Nadler,*Abramov's formula for reparametrizations of n-flows*(to appear).**[O]**D. Ornstein,*Randomness, ergodic theory, and dynamical systems*, Yale Mathematical Monographs, No. 5.**[P-S]**Charles Pugh and Michael Shub,*Ergodic elements of ergodic actions*, Compositio Math.**23**(1971), 115–122. MR**0283174****[R1]**D. Rudolph,*Nonequivalence of measure-preserving transformations*, notes.**[R2]**-,*A dye theorem for n-flows*, (unpublished).**[R3]**-,*An integrably Lipschitz reparametrization of an n-flow is isomorphic to some tempered reparametrization of the same n-flow*(unpublished).**[Sa]**E. Satayev,*An invariant of monotone equivalence which determines families of automorphisms which are monotone equivalent to a Bernoulli automorphism*, Proc. Fourth Sympos. on Information Theory, Part I, Moscow-Leningrad, 1976. (Russian)**[S]**Ja. G. Sinaĭ,*A weak isomorphism of transformations with invariant measure*, Dokl. Akad. Nauk SSSR**147**(1962), 797–800 (Russian). MR**0161960**

Ja. G. Sinaĭ,*On a weak isomorphism of transformations with invariant measure*, Mat. Sb. (N.S.)**63 (105)**(1964), 23–42 (Russian). MR**0161961****[W]**B. Weiss,*Equivalence of measure preserving transformations*, Lecture Notes, Hebrew University of Jerusalem.

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DOI:
https://doi.org/10.1090/S0002-9947-1979-0546919-6

Article copyright:
© Copyright 1979
American Mathematical Society