The structure of hyperfinite Borel equivalence relations
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- by R. Dougherty, S. Jackson and A. S. Kechris PDF
- Trans. Amer. Math. Soc. 341 (1994), 193-225 Request permission
Abstract:
We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such equivalence relations. In particular, since the only possible such cardinalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorphism types. Canonical examples of each type are also discussed.References
- Scot Adams, An equivalence relation that is not freely generated, Proc. Amer. Math. Soc. 102 (1988), no. 3, 565–566. MR 928981, DOI 10.1090/S0002-9939-1988-0928981-2
- Pradeep Chaube and M. G. Nadkarni, A version of Dye’s theorem for descriptive dynamical systems, Sankhyā Ser. A 49 (1987), no. 3, 288–304. MR 1056022
- P. Chaube and M. G. Nadkarni, On orbit equivalence of Borel automorphisms, Proc. Indian Acad. Sci. Math. Sci. 99 (1989), no. 3, 255–261. MR 1032712, DOI 10.1007/BF02864398
- A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems 1 (1981), no. 4, 431–450 (1982). MR 662736, DOI 10.1017/s014338570000136x
- H. A. Dye, On groups of measure preserving transformations. I, Amer. J. Math. 81 (1959), 119–159. MR 131516, DOI 10.2307/2372852
- Edward G. Effros, Transformation groups and $C^{\ast }$-algebras, Ann. of Math. (2) 81 (1965), 38–55. MR 174987, DOI 10.2307/1970381
- Edward G. Effros, Polish transformation groups and classification problems, General topology and modern analysis (Proc. Conf., Univ. California, Riverside, Calif., 1980) Academic Press, New York-London, 1981, pp. 217–227. MR 619045
- Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 578656, DOI 10.1090/S0002-9947-1977-0578656-4
- Nathaniel A. Friedman, Introduction to ergodic theory, Van Nostrand Reinhold Mathematical Studies, No. 29, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1970. MR 0435350
- Colin C. Graham and O. Carruth McGehee, Essays in commutative harmonic analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 238, Springer-Verlag, New York-Berlin, 1979. MR 550606
- Toshihiro Hamachi and Motosige Osikawa, Ergodic groups of automorphisms and Krieger’s theorems, Seminar on Mathematical Sciences, vol. 3, Keio University, Department of Mathematics, Yokohama, 1981. MR 617740
- L. A. Harrington, A. S. Kechris, and A. Louveau, A Glimm-Effros dichotomy for Borel equivalence relations, J. Amer. Math. Soc. 3 (1990), no. 4, 903–928. MR 1057041, DOI 10.1090/S0894-0347-1990-1057041-5
- Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Graduate Texts in Mathematics, No. 25, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing. MR 0367121
- Witold Hurewicz, Ergodic theorem without invariant measure, Ann. of Math. (2) 45 (1944), 192–206. MR 9427, DOI 10.2307/1969081
- Yitzhak Katznelson and Benjamin Weiss, The construction of quasi-invariant measures, Israel J. Math. 12 (1972), 1–4. MR 316679, DOI 10.1007/BF02764806
- Alexander S. Kechris, Amenable equivalence relations and Turing degrees, J. Symbolic Logic 56 (1991), no. 1, 182–194. MR 1131739, DOI 10.2307/2274913
- Alexander S. Kechris, The structure of Borel equivalence relations in Polish spaces, Set theory of the continuum (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 26, Springer, New York, 1992, pp. 89–102. MR 1233813, DOI 10.1007/978-1-4613-9754-0_{7}
- Alexander S. Kechris, Countable sections for locally compact group actions, Ergodic Theory Dynam. Systems 12 (1992), no. 2, 283–295. MR 1176624, DOI 10.1017/S0143385700006751
- Ju. I. Kifer and S. A. Pirogov, The decomposition of quasi-invariant measures into ergodic components, Uspehi Mat. Nauk 27 (1972), no. 5(167), 239–240 (Russian). MR 0419733
- Wolfgang Krieger, On non-singular transformations of a measure space. I, II, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 11 (1969), 83–97; ibid. 11 (1969), 98–119. MR 240279, DOI 10.1007/BF00531811
- Wolfgang Krieger, On quasi-invariant measures in uniquely ergodic systems, Invent. Math. 14 (1971), 184–196. MR 293060, DOI 10.1007/BF01418888
- Wolfgang Krieger, On ergodic flows and the isomorphism of factors, Math. Ann. 223 (1976), no. 1, 19–70. MR 415341, DOI 10.1007/BF01360278
- Gustave Choquet, Topology, Pure and Applied Mathematics, Vol. XIX, Academic Press, New York-London, 1966. Translated from the French by Amiel Feinstein. MR 0193605
- R. Daniel Mauldin and S. M. Ulam, Mathematical problems and games, Adv. in Appl. Math. 8 (1987), no. 3, 281–344. MR 898709, DOI 10.1016/0196-8858(87)90026-1
- Calvin C. Moore, Ergodic theory and von Neumann algebras, Operator algebras and applications, Part 2 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 179–226. MR 679505
- Yiannis N. Moschovakis, Descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 561709
- M. G. Nadkarni, Descriptive ergodic theory, Measure and measurable dynamics (Rochester, NY, 1987) Contemp. Math., vol. 94, Amer. Math. Soc., Providence, RI, 1989, pp. 191–206. MR 1012990, DOI 10.1090/conm/094/1012990
- M. G. Nadkarni, On the existence of a finite invariant measure, Proc. Indian Acad. Sci. Math. Sci. 100 (1990), no. 3, 203–220. MR 1081705 —, Orbit equivalence and Kakutani equivalence in descriptive setting,, reprint, 1991.
- Klaus Schmidt, Cocycles on ergodic transformation groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan Co. of India, Ltd., Delhi, 1977. MR 0578731
- Klaus Schmidt, Algebraic ideas in ergodic theory, CBMS Regional Conference Series in Mathematics, vol. 76, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1074576, DOI 10.1090/cbms/076
- Theodore A. Slaman and John R. Steel, Definable functions on degrees, Cabal Seminar 81–85, Lecture Notes in Math., vol. 1333, Springer, Berlin, 1988, pp. 37–55. MR 960895, DOI 10.1007/BFb0084969
- Dennis Sullivan, B. Weiss, and J. D. Maitland Wright, Generic dynamics and monotone complete $C^\ast$-algebras, Trans. Amer. Math. Soc. 295 (1986), no. 2, 795–809. MR 833710, DOI 10.1090/S0002-9947-1986-0833710-X C. Sutherland, Orbit equivalence: lectures on Krieger’s theorem, Univ. of Oslo Lecture Note Series, no. 23, 1976.
- V. S. Varadarajan, Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc. 109 (1963), 191–220. MR 159923, DOI 10.1090/S0002-9947-1963-0159923-5 A. M. Vershik, The action of $PSL(2,\mathbb {R})$ on ${P_1}\mathbb {R}$ is approximable, Russian Math. Surveys (1) 33 (1978), 221-222. —, Trajectory theory, Dynamical Systems. II (Ya. G. Sinai, eds.), Springer-Verlag, 1989, pp. 77-98.
- V. M. Wagh, A descriptive version of Ambrose’s representation theorem for flows, Proc. Indian Acad. Sci. Math. Sci. 98 (1988), no. 2-3, 101–108. MR 994127, DOI 10.1007/BF02863630
- Benjamin Weiss, Orbit equivalence of nonsingular actions, Ergodic theory (Sem., Les Plans-sur-Bex, 1980) Monograph. Enseign. Math., vol. 29, Univ. Genève, Geneva, 1981, pp. 77–107. MR 609897
- Benjamin Weiss, Measurable dynamics, Conference in modern analysis and probability (New Haven, Conn., 1982) Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 395–421. MR 737417, DOI 10.1090/conm/026/737417 —, Countable generators in dynamics—universal minimal models, Measure and Measurable Dynamics (R. D. Mauldin et al., eds.), Contemp. Math., vol. 94, Amer. Math. Soc., Providence, RI, 1989, pp. 321-326.
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 341 (1994), 193-225
- MSC: Primary 03E15; Secondary 03H05, 28A05, 28D99
- DOI: https://doi.org/10.1090/S0002-9947-1994-1149121-0
- MathSciNet review: 1149121