Hostname: page-component-7c8c6479df-5xszh Total loading time: 0 Render date: 2024-03-29T02:04:09.962Z Has data issue: false hasContentIssue false

Speedups of ergodic group extensions

Published online by Cambridge University Press:  01 May 2012

ANDREY BABICHEV
Affiliation:
Wesleyan University, Middletown, CT, USA (email: ababichev@wesleyan.edu)
ROBERT M. BURTON
Affiliation:
Oregon State University, Corvallis, OR, USA (email: dr.bob.math@gmail.com)
ADAM FIELDSTEEL
Affiliation:
Wesleyan University, Middletown, CT, USA (email: afieldsteel@wesleyan.edu)

Abstract

We prove that for all ergodic extensions $S_{1}$ of a transformation by a locally compact second countable group $G$, and for all $G$-extensions $ S_{2} $ of an aperiodic transformation, there is a relative speedup of $ S_{1} $ that is relatively isomorphic to $S_{2}$. We apply this result to give necessary and sufficient conditions for two ergodic $n$-point or countable extensions to be related in this way.

Type
Research Article
Copyright
Copyright © 2012 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[AOW]Arnoux, P., Ornstein, D. S. and Weiss, B.. Cutting and stacking, interval exchanges and geometric models. Israel J. Math. 50(1–2) (1985), 160168.CrossRefGoogle Scholar
[BF]Babichev, A. and Fieldsteel, A.. Speedups of compact group extensions, arXiv:1112.4377.Google Scholar
[B]Becker, H.. Polish group actions: dichotomies and generalized elementary embeddings. J. Amer. Math. Soc. 11(2) (1998), 397449.CrossRefGoogle Scholar
[F]Fieldsteel, A.. Factor orbit equivalence of compact group extensions. Israel J. Math. 38(4) (1981), 289303.Google Scholar
[G]Gerber, M.. Factor orbit equivalence of compact group extensions and classification of finite extensions of ergodic automorphisms. Israel J. Math. 57 (1987), 2848.Google Scholar
[GS]Golodets, V. Ya. and Sinel’shchikov, S. D.. Classification and structure of cocycles of amenable ergodic equivalence relations. J. Funct. Anal. 121 (1994), 455485.Google Scholar
[H]Herman, M.. Construction de difféomorphismes ergodiques. Unpublished manuscript, 1979.Google Scholar
[KR1]Kammeyer, J. and Rudolph, D. J.. Restricted orbit equivalence for ergodic $\mathbb {Z}^{d}$ actions, I. Ergod. Th. & Dynam. Sys. 17(5) (1997), 10831129.Google Scholar
[KR2]Kammeyer, J. and Rudolph, D. J.. Restricted Orbit Equivalence for Actions of Discrete Amenable Groups (Cambridge Tracts in Mathematics, 146). Cambridge University Press, Cambridge, 2002.Google Scholar
[N]Neveu, J.. Une démonstration simplifée et une extension de la formule d’Abramov sur l’entropie des transformations induites. Z. Wahrscheinlichkeitst. Verw. Geb. 13 (1969), 135140.Google Scholar
[O]Ornstein, D. S.. Ergodic Theory, Randomness and Dynamical Systems. Yale University Press, New Haven, CT, 1970.Google Scholar
[OW]Ornstein, D. S. and Weiss, B.. Any flow is an orbit factor of any flow. Ergod. Th. & Dynam. Sys. 4 (1984), 105116.Google Scholar
[R]Rudolph, D. J.. Restricted orbit equivalence. Mem. Amer. Math. Soc. 323 (1985), 149pp.Google Scholar
[Z]Zimmer, R.. Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Funct. Anal. 27 (1978), 350372.Google Scholar