Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Reparametrization of $ n$-flows of zero entropy

Authors: J. Feldman and D. Nadler
Journal: Trans. Amer. Math. Soc. 256 (1979), 289-304
MSC: Primary 28D15
Correction: Trans. Amer. Math. Soc. 264 (1981), 583-585.
MathSciNet review: 546919
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \phi $, $ \psi $ be two ergodic n-parameter flows which preserve finite probability measures on their spaces X, Y. Let T be a nullset-preserving map: $ X\, \to \,Y$ sending each $ \phi $-orbit homeomorphically to a $ \phi $-orbit. Then $ \phi $, $ \psi $ are called homeomorphically orbit-equivalent. For $ n\, = \,1$, 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 $ n\, = \,1$. In this paper we carry out this program, but only for the case of zero entropy.

References [Enhancements On Off] (What's this?)

  • [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,
  • [F1] J. Feldman, New 𝐾-automorphisms and a problem of Kakutani, Israel J. Math. 24 (1976), no. 1, 16–38. MR 0409763,
  • [F2] -, r-entropy, equipartition, and Ornstein's isomorphism theorem in $ {{\textbf{R}}^n}$ (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,
  • [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, $ n\, > \,1$ (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.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 28D15

Retrieve articles in all journals with MSC: 28D15

Additional Information

Article copyright: © Copyright 1979 American Mathematical Society