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Lebesgue orbit equivalence of multidimensional Borel flows: A picturebook of tilings

Published online by Cambridge University Press:  08 March 2016

KONSTANTIN SLUTSKY
KONSTANTIN SLUTSKY*
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street, Chicago, IL 60607-7045, USA email kslutsky@gmail.com
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Abstract

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The main result of the paper is classification of free multidimensional Borel flows up to Lebesgue orbit equivalence, by which we mean an orbit equivalence that preserves the Lebesgue measure on each orbit. Two non-smooth $\mathbb{R}^{d}$-flows are shown to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 6 , September 2017 , pp. 1966 - 1996
Copyright
© Cambridge University Press, 2016 

References

Becker, H. and Kechris, A. S.. The Descriptive Set Theory of Polish Group Actions (London Mathematical Society Lecture Note Series, 232) . Cambridge University Press, Cambridge, 1996.CrossRefGoogle Scholar
Dougherty, R., Jackson, S. and Kechris, A. S.. The structure of hyperfinite borel equivalence relations. Trans. Amer. Math. Soc. 341(1) (1994), 193–225.Google Scholar
Feldman, J. and Moore, C. C.. Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Amer. Math. Soc. 234(2) (1977), 289–324.CrossRefGoogle Scholar
Gao, S. and Jackson, S.. Countable abelian group actions and hyperfinite equivalence relations. Invent. Math. 201(1) (2015), 309–383.Google Scholar
Jackson, S., Kechris, A. S. and Louveau, A.. Countable Borel equivalence relations. J. Math. Log. 2(1) (2002), 1–80.Google Scholar
Kechris, A. S.. Countable sections for locally compact group actions. Ergod. Th. & Dynam. Sys. 12(2) (1992), 283–295.CrossRefGoogle Scholar
Kechris, A. S.. Classical Descriptive Set Theory (Graduate Texts in Mathematics, 156) . Springer, New York, 1995.CrossRefGoogle Scholar
Kyed, D., Petersen, H. D. and Vaes, S.. L 2 -betti numbers of locally compact groups and their cross section equivalence relations. Trans. Amer. Math. Soc. 367(7) (2015), 4917–4956.Google Scholar
Krengel, U.. On Rudolph’s representation of aperiodic flows. Ann. Inst. H. Poincaré Sect. B (N.S.) 12(4) (1976), 319–338.Google Scholar
Kechris, A. S., Solecki, S. and Todorčević, S.. Borel chromatic numbers. Adv. Math. 141(1) (1999), 1–44.Google Scholar
Lind, D. A.. Locally compact measure preserving flows. Adv. Math. 15 (1975), 175–193.Google Scholar
Miller, B. D. and Rosendal, C.. Descriptive Kakutani equivalence. J. Eur. Math. Soc. (JEMS) 12(1) (2010), 179–219.Google Scholar
Nadkarni, M. G.. On the existence of a finite invariant measure. Proc. Indian Acad. Sci. Math. Sci. 100(3) (1990), 203–220.Google Scholar
Rudolph, D. J.. Smooth orbit equivalence of ergodic ℝ d actions, d ≥ 2. Trans. Amer. Math. Soc. 253 (1979), 291–302.Google Scholar
Rudolph, D. J.. Rectangular tilings of ℝ n and free ℝ n actions. Dynamical Systems (College Park, MD) 1986–1987 (Lecture Notes in Mathematics, 1342) . Springer, Berlin, 1988, pp. 653–688.Google Scholar
Struble, R. A.. Metrics in locally compact groups. Compositio Math. 28 (1974), 217–222.Google Scholar