Reparametrization of $n$-flows of zero entropy
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- by J. Feldman and D. Nadler PDF
- Trans. Amer. Math. Soc. 256 (1979), 289-304 Request permission
Correction: Trans. Amer. Math. Soc. 264 (1981), 583-585.
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
- L. M. Abramov, The entropy of a derived automorphism, Dokl. Akad. Nauk SSSR 128 (1959), 647–650 (Russian). MR 0113984
- H. A. Dye, On groups of measure preserving transformations. I, Amer. J. Math. 81 (1959), 119–159. MR 131516, DOI 10.2307/2372852
- J. Feldman, New $K$-automorphisms and a problem of Kakutani, Israel J. Math. 24 (1976), no. 1, 16–38. MR 409763, DOI 10.1007/BF02761426 —, r-entropy, equipartition, and Ornstein’s isomorphism theorem in ${{\textbf {R}}^n}$ (to appear).
- Shizuo Kakutani, Induced measure preserving transformations, Proc. Imp. Acad. Tokyo 19 (1943), 635–641. MR 14222
- A. B. Katok, Time change, monotone equivalence, and standard dynamical systems, Dokl. Akad. Nauk SSSR 223 (1975), no. 4, 789–792 (Russian). MR 0412383
- D. A. Lind, Locally compact measure preserving flows, Advances in Math. 15 (1975), 175–193. MR 382595, DOI 10.1016/0001-8708(75)90133-4 D. Nadler, Abramov’s formula for reparametrizations of n-flows (to appear). D. Ornstein, Randomness, ergodic theory, and dynamical systems, Yale Mathematical Monographs, No. 5.
- Charles Pugh and Michael Shub, Ergodic elements of ergodic actions, Compositio Math. 23 (1971), 115–122. MR 283174 D. Rudolph, Nonequivalence of measure-preserving transformations, notes. —, A dye theorem for n-flows, $n\, > \,1$ (unpublished). —, An integrably Lipschitz reparametrization of an n-flow is isomorphic to some tempered reparametrization of the same n-flow (unpublished). 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)
- Ja. G. Sinaĭ, A weak isomorphism of transformations with invariant measure, Dokl. Akad. Nauk SSSR 147 (1962), 797–800 (Russian). MR 0161960 B. Weiss, Equivalence of measure preserving transformations, Lecture Notes, Hebrew University of Jerusalem.
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 256 (1979), 289-304
- MSC: Primary 28D15
- DOI: https://doi.org/10.1090/S0002-9947-1979-0546919-6
- MathSciNet review: 546919