Encyclopedia of Complexity and Systems Science

2009 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Ergodic Theory: Non-singular Transformations

  • Alexandre I. Danilenko
  • Cesar E. Silva
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_183

Definition of the Subject

An abstract measurable dynamical system consists of a set X (phase space) with a transformation\( { T \colon X \to X } \)

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  1. 1.
    Aaronson J (1983) The eigenvalues of nonsingular transformations. IsrJ Math 45:297–312MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aaronson J (1987) The intrinsic normalizing constants of transformationspreserving infinite measures. J Analyse Math 49:239–270MathSciNetzbMATHGoogle Scholar
  3. 3.
    Aaronson J (1997) An Introduction to Infinite Ergodic Theory. Amer Math Soc,ProvidencezbMATHGoogle Scholar
  4. 4.
    Aaronson J, Lemańczyk M (2005) Exactness of Rokhlin endomorphisms and weak mixing of Poisson boundaries, Algebraic and Topological Dynamics. Contemporary Mathematics, vol 385. Amer Math Soc, Providence 77–88Google Scholar
  5. 5.
    Aaronson J, Lin M, Weiss B (1979) Mixing properties of Markov operators andergodic transformations, and ergodicity of cartesian products. Isr J Math 33:198–224MathSciNetzbMATHGoogle Scholar
  6. 6.
    Aaronson J, Nadkarni M (1987) \( { L_\infty } \) eigenvalues and L 2spectra of nonsingular transformations. Proc Lond Math Soc 55(3):538–570MathSciNetzbMATHGoogle Scholar
  7. 7.
    Aaronson J, Nakada H (2000) Multiple recurrence of Markov shifts and otherinfinite measure preserving transformations. Isr J Math 117:285–310MathSciNetzbMATHGoogle Scholar
  8. 8.
    Aaronson J, Park KK (2008) Predictability, entropy and information of infinite transformations, preprint. ArXiv:0705:2148v3Google Scholar
  9. 9.
    Aaronson J, Weiss B (2004) On Herman's theorem for ergodic, amenable groupextensions of endomorphisms. Ergod Theory Dynam Syst 5:1283–1293MathSciNetGoogle Scholar
  10. 10.
    Adams S, Elliott GA, Giordano T (1994) Amenable actions of groups. Trans AmerMath Soc 344:803–822MathSciNetzbMATHGoogle Scholar
  11. 11.
    Adams T, Friedman N, Silva CE (1997) Rank-One Weak Mixing for NonsingularTransformations. Isr J Math 102:269–281MathSciNetzbMATHGoogle Scholar
  12. 12.
    Adams T, Friedman N, Silva CE (2001) Rank one power weak mixing fornonsingular transformations. Ergod Theory Dynam Systems 21:1321–1332MathSciNetzbMATHGoogle Scholar
  13. 13.
    Ageev ON, Silva CE (2002) Genericity of rigid and multiply recurrent infinitemeasure‐preserving and nonsingular transformations. In: Proceedings of the 16th Summer Conference on General Topology and its Applications. TopologyProc 26(2):357–365MathSciNetGoogle Scholar
  14. 14.
    Albeverio S, Hoegh-Krohn R, Testard D, Vershik AM (1983) Factorialrepresentations of Path groups. J Funct Anal 51:115–231MathSciNetzbMATHGoogle Scholar
  15. 15.
    Atkinson G (1976) Recurrence of co‐cycles and random walks. J LondMath Soc 13:486–488zbMATHGoogle Scholar
  16. 16.
    Babillot M, Ledrappier F (1998) Geodesic paths and horocycle flow on abelian covers. In: Lie groups and ergodic theory. Tata Inst Fund Res Stud Math 14, Tata Inst Fund Res, Bombay, pp 1–32Google Scholar
  17. 17.
    Bergelson V, Leibman A (1996) Polynomial extensions of van der Waerden's andSemerédi's theorems. J Amer Math Soc 9:725–753MathSciNetzbMATHGoogle Scholar
  18. 18.
    Bezuglyi SI, Golodets VY (1985) Groups of measure space transformations andinvariants of outer conjugation for automorphisms from normalizers of type III full groups. J Funct Anal60(3):341–369MathSciNetzbMATHGoogle Scholar
  19. 19.
    Bezuglyi SI, Golodets VY (1991) Weak equivalence and the structures ofcocycles of an ergodic automorphism. Publ Res Inst Math Sci 27(4):577–625MathSciNetGoogle Scholar
  20. 20.
    Bowles A, Fidkowski L, Marinello A, Silva CE (2001) Double ergodicity ofnonsingular transformations and infinite measure‐preserving staircase transformations. Ill J Math45(3):999–1019MathSciNetzbMATHGoogle Scholar
  21. 21.
    Chacon RV, Friedman NA (1965) Approximation and invariant measures. ZWahrscheinlichkeitstheorie Verw Gebiete 3:286–295Google Scholar
  22. 22.
    Choksi JR, Eigen S, Prasad V (1989) Ergodic theory on homogeneous measure algebras revisited. In: Mauldin RD, Shortt RM, Silva CE (eds) Measure and measurable dynamics. Contemp Math 94, Amer Math Soc, Providence, pp 73–85Google Scholar
  23. 23.
    Choksi JR, Hawkins JM, Prasad VS (1987) Abelian cocylces for nonsingularergodic transformations and the genericity of type \( { \mathrm{III}_1 } \) transformations. Monat fur Math 103:187–205MathSciNetzbMATHGoogle Scholar
  24. 24.
    Choksi JR, Kakutani S (1979) Residuality of ergodic measurable transformationsand of ergodic transformations which preserve an infinite measure. Ind Univ Math J 28:453–469MathSciNetzbMATHGoogle Scholar
  25. 25.
    Choksi JR, Nadkarni MG (1994) The maximal spectral type of a rank onetransformation. Canad Math Bull 37(1):29–36MathSciNetzbMATHGoogle Scholar
  26. 26.
    Choksi JR, Nadkarni MG (2000) Genericity of nonsingular transformations withinfinite ergodic index. Colloq Math 84/85:195–201MathSciNetGoogle Scholar
  27. 27.
    Choksi JR, Prasad VS (1983) Approximation and Baire category theorems in ergodic theory. In: Belley JM, Dubois J, Morales P (eds) Measure theory and its applications. Lect Notes Math 1033. Springer, Berlin, pp 94–113Google Scholar
  28. 28.
    Connes A (1975) On the hierarchy of W Krieger. Ill J Math19:428–432MathSciNetzbMATHGoogle Scholar
  29. 29.
    Connes A, Feldman J, Weiss B (1981) An amenable equivalence relation isgenerated by a single transformation. Ergod Theory Dynam Systems 1:431–450MathSciNetzbMATHGoogle Scholar
  30. 30.
    Connes A, Krieger W (1977) Measure space automorphisms, the normalizers oftheir full groups, and approximate finiteness. J Funct Anal 24(4):336–352MathSciNetzbMATHGoogle Scholar
  31. 31.
    Connes A, Woods EJ (1985) Approximately transitive flows and ITPFIfactors. Ergod Theory Dynam Syst 5(2):203–236MathSciNetzbMATHGoogle Scholar
  32. 32.
    Connes A, Woods EJ (1989) Hyperfinite von Neumann algebras and Poissonboundaries of time dependent random walks. Pac J Math 37:225–243MathSciNetGoogle Scholar
  33. 33.
    Cornfeld IP, Fomin VS, Sinaĭ YG (1982) Ergodic theory. Grundlehren der Mathematischen Wissenschaften, vol 245. Springer, New YorkGoogle Scholar
  34. 34.
    Danilenko AI (1995) The topological structure of Polish groups and groupoidsof measure space transformations. Publ Res Inst Math Sci 31(5):913–940MathSciNetzbMATHGoogle Scholar
  35. 35.
    Danilenko AI (1998) Quasinormal subrelations of ergodic equivalencerelations. Proc Amer Math Soc 126(11):3361–3370MathSciNetzbMATHGoogle Scholar
  36. 36.
    Danilenko AI (2001) Funny rank one weak mixing for nonsingular Abelianactions. Isr J Math 121:29–54MathSciNetzbMATHGoogle Scholar
  37. 37.
    Danilenko AI (2001) Strong orbit equivalence of locally compact Cantor minimalsystems. Int J Math 12:113–123MathSciNetzbMATHGoogle Scholar
  38. 38.
    Danilenko AI (2004) Infinite rank one actions and nonsingular Chacontransformations. Ill J Math 48(3):769–786MathSciNetzbMATHGoogle Scholar
  39. 39.
    Danilenko AI (2007) On simplicity concepts for ergodic actions. J d'AnalMath 102:77–118Google Scholar
  40. 40.
    Danilenko AI (2007) \( { (C,F) } \)‑actions in ergodic theory. In: Kapranov M, Kolyada S, Manin YI, Moree P, Potyagailo L (eds) Geometry and Dynamics of Groups and Spaces. Progr Math 265:325–351MathSciNetGoogle Scholar
  41. 41.
    Danilenko AI, Golodets YV (1996) On extension of cocycles to normalizerelements, outer conjugacy, and related problems. Trans Amer Math Soc 348(12):4857–4882MathSciNetzbMATHGoogle Scholar
  42. 42.
    Danilenko AI, Hamachi T (2000) On measure theoretical analogues of theTakesaki structure theorem for type III factors. Colloq Math 84/85:485–493MathSciNetGoogle Scholar
  43. 43.
    Danilenko AI, Lemańczyk M (2005) A class of multipliers for\( { \mathcal{W}^\perp } \). Isr J Math148:137–168Google Scholar
  44. 44.
    Danilenko AI, Rudolph DJ: Conditional entropy theory in infinite measure and a question of Krengel. Isr J Math, to appearGoogle Scholar
  45. 45.
    Danilenko AI, Silva CE (2004) Multiple and polynomial recurrence for Abelianactions in infinite measure. J Lond Math Soc 2 69(1):183–200MathSciNetGoogle Scholar
  46. 46.
    Danilenko AI, Solomko AV: Infinite measure preserving flows with infinite ergodic index. Colloq Math, to appearGoogle Scholar
  47. 47.
    Day S, Grivna B, McCartney E, Silva CE (1999) Power Weakly Mixing InfiniteTransformations. N Y J Math 5:17–24MathSciNetzbMATHGoogle Scholar
  48. 48.
    del Junco A (1978) A simple measure‐preserving transformationwith trivial centralizer. Pac J Math 79:357–362Google Scholar
  49. 49.
    del Junco A, Rudolph DJ (1987) On ergodic actions whose self‐joiningsare graphs. Ergod Theory Dynam Syst 7:531–557zbMATHGoogle Scholar
  50. 50.
    del Junco A, Silva CE (1995) Prime type \( { \mathrm{III}_\lambda } \) automorphisms: An instance ofcoding techniques applied to nonsingular maps. In: Takahashi Y (ed) Fractals and Dynamics. Plenum, NewYork, pp 101–115Google Scholar
  51. 51.
    del Junco A, Silva CE (2003) On factors of nonsingular Cartesianproducts. Ergod Theory Dynam Syst 23(5):1445–1465zbMATHGoogle Scholar
  52. 52.
    Derriennic Y, Frączek K, Lemańczyk M, Parreau F (2008) Ergodicautomorphisms whose weak closure of off‐diagonal measures consists of ergodic self‐joinings. Colloq Math110:81–115Google Scholar
  53. 53.
    Dixmier J (1969) Les \( { C^* } \)-algèbres et leurs représentations. Gauthier‐Villars Editeur,ParisGoogle Scholar
  54. 54.
    Dooley AH, Hamachi T (2003) Nonsingular dynamical systems, Bratteli diagramsand Markov odometers. Isr J Math 138:93–123MathSciNetzbMATHGoogle Scholar
  55. 55.
    Dooley AH, Hamachi T (2003) Markov odometer actions not of product type. ErgodTheory Dynam Syst 23(3):813–829MathSciNetzbMATHGoogle Scholar
  56. 56.
    Dooley AH, Klemes I, Quas AN (1998) Product and Markov measures oftype III. J Aust Math Soc Ser A 65(1):84–110MathSciNetzbMATHGoogle Scholar
  57. 57.
    Dooley AH, Mortiss G: On the critical dimension of product odometers, preprintGoogle Scholar
  58. 58.
    Dooley AH, Mortiss G (2006) On the critical dimension and AC entropy forMarkov odometers. Monatsh Math 149:193–213MathSciNetzbMATHGoogle Scholar
  59. 59.
    Dooley AH, Mortiss G (2007) The critical dimensions of Hamachi shifts. TohokuMath J 59(2):57–66MathSciNetzbMATHGoogle Scholar
  60. 60.
    Dye H (1963) On groups of measure‐preserving transformations I. AmerJ Math 81:119–159, and II, Amer J Math 85:551–576MathSciNetGoogle Scholar
  61. 61.
    Effros EG (1965) Transformation groups and \( { C^* } \)-algebras. Ann Math81(2):38–55MathSciNetzbMATHGoogle Scholar
  62. 62.
    Eigen SJ (1981) On the simplicity of the full group of ergodictransformations. Isr J Math 40(3–4):345–349MathSciNetzbMATHGoogle Scholar
  63. 63.
    Eigen SJ (1982) The group of measure preserving transformations of [0,1] hasno outer automorphisms. Math Ann 259:259–270MathSciNetzbMATHGoogle Scholar
  64. 64.
    Eigen S, Hajian A, Halverson K (1998) Multiple recurrence and infinite measurepreserving odometers. Isr J Math 108:37–44MathSciNetzbMATHGoogle Scholar
  65. 65.
    Fedorov A (1985) Krieger's theorem for cocycles, preprintGoogle Scholar
  66. 66.
    Feldman J, Moore CC (1977) Ergodic equivalence relations, cohomology, and vonNeumann algebras. I. Trans Amer Math Soc 234:289–324MathSciNetzbMATHGoogle Scholar
  67. 67.
    Ferenczi S (1985) Systèmes de rang un gauche. Ann Inst H Poincaré ProbabStatist 21(2):177–186MathSciNetzbMATHGoogle Scholar
  68. 68.
    Friedman NA (1970) Introduction to Ergodic Theory. Van Nostrand Reinhold Mathematical Studies, No 29. Van Nostrand Reinhold Co., New YorkGoogle Scholar
  69. 69.
    Furstenberg H (1967) Disjointness in ergodic theory, minimal sets anddiophantine approximation. Math Syst Theory 1:1–49MathSciNetzbMATHGoogle Scholar
  70. 70.
    Furstenberg H (1981) Recurrence in Ergodic Theory and Combinatorial NumberTheory. Princeton University Press, PrincetonGoogle Scholar
  71. 71.
    Furstenberg H, Weiss B (1978) The finite multipliers of infinite ergodic transformations, The structure of attractors in dynamical systems. In: Markley NG, Martin JC, Perrizo W (eds) Lecture Notes in Math 668. Springer, Berlin, pp 127–132Google Scholar
  72. 72.
    Gårding L, Wightman AS (1954) Representation of anticommutationrelations. Proc Nat Acad Sci USA 40:617–621Google Scholar
  73. 73.
    Giordano T, Skandalis G (1985) Krieger factors isomorphic to their tensorsquare and pure point spectrum flows. J Funct Anal 64(2):209–226MathSciNetzbMATHGoogle Scholar
  74. 74.
    Giordano T, Skandalis G (1985) On infinite tensor products of factors of typeI 2. Ergod Theory Dynam Syst 5:565–586MathSciNetzbMATHGoogle Scholar
  75. 75.
    Glasner E (1994) On the multipliers of \( { \mathcal{W}^\perp } \). Ergod Theory Dynam Syst14:129–140MathSciNetzbMATHGoogle Scholar
  76. 76.
    Glimm J (1961) Locally compact transformation groups. Trans Amer Math Soc101:124–138MathSciNetzbMATHGoogle Scholar
  77. 77.
    Golodets YV (1969) A description of the representations ofanticommutation relations. Uspehi Matemat Nauk 24(4):43–64Google Scholar
  78. 78.
    Golodets YV, Sinel'shchikov SD (1983) Existence and uniqueness of cocycles ofergodic automorphism with dense range in amenable groups. Preprint FTINT AN USSR, pp 19–83Google Scholar
  79. 79.
    Golodets YV, Sinel'shchikov SD (1985) Locally compact groups appearing asranges of cocycles of ergodic Z‑actions. Ergod Theory Dynam Syst5:47–57MathSciNetzbMATHGoogle Scholar
  80. 80.
    Golodets YV, Sinel'shchikov SD (1990) Amenable ergodic actions of groups andimages of cocycles. Dokl Akad Nauk SSSR 312(6):1296–1299, in RussianMathSciNetGoogle Scholar
  81. 81.
    Golodets YV, Sinel'shchikov SD (1994) Classification and structure of cocyclesof amenable ergodic equivalence relations. J Funct Anal 121(2):455–485MathSciNetzbMATHGoogle Scholar
  82. 82.
    Gruher K, Hines F, Patel D, Silva CE, Waelder R (2003) Power weak mixing doesnot imply multiple recurrence in infinite measure and other counterexamples. N Y J Math 9:1–22MathSciNetzbMATHGoogle Scholar
  83. 83.
    Hajian AB, Kakutani S (1964) Weakly wandering sets and invariantmeasures. Trans Amer Math Soc 110:136–151MathSciNetzbMATHGoogle Scholar
  84. 84.
    Halmos PR (1946) An ergodic theorem. Proc Nat Acad Sci USA32:156–161MathSciNetADSzbMATHGoogle Scholar
  85. 85.
    Halmos PR (1956) Lectures on ergodic theory. Publ Math Soc Jpn 3Google Scholar
  86. 86.
    Hamachi T (1981) The normalizer group of an ergodic automorphism oftype III and the commutant of an ergodic flow. J Funct Anal 40:387–403MathSciNetzbMATHGoogle Scholar
  87. 87.
    Hamachi T (1981) On a Bernoulli shift with nonidentical factor measures. ErgodTheory Dynam Syst 1:273–283MathSciNetzbMATHGoogle Scholar
  88. 88.
    Hamachi T (1992) A measure theoretical proof of the Connes–Woodstheorem on AT-flows. Pac J Math 154:67–85MathSciNetzbMATHGoogle Scholar
  89. 89.
    Hamachi T, Kosaki H (1993) Orbital factor map. Ergod Theory Dynam Syst13:515–532MathSciNetzbMATHGoogle Scholar
  90. 90.
    Hamachi T, Osikawa M (1981) Ergodic groups of automorphisms and Krieger'stheorems. Semin Math Sci 3, Keio UnivGoogle Scholar
  91. 91.
    Hamachi T, Osikawa M (1986) Computation of the associated flows of \( { \mathrm{ITPFI}_2 } \) factors of type \( { \mathrm{III}_0 } \). In: Geometric methods in operator algebras. Pitman Res Notes Math Ser 123, Longman Sci Tech, Harlow, pp 196–210Google Scholar
  92. 92.
    Hamachi T, Silva CE (2000) On nonsingular Chacon transformations. IllJ Math 44:868–883MathSciNetzbMATHGoogle Scholar
  93. 93.
    Hawkins JM (1982) Non-ITPFI diffeomorphisms. Isr J Math42:117–131MathSciNetzbMATHGoogle Scholar
  94. 94.
    Hawkins JM (1983) Smooth type III diffeomorphisms of manifolds. TransAmer Math Soc 276:625–643MathSciNetzbMATHGoogle Scholar
  95. 95.
    Hawkins JM (1990) Diffeomorphisms of manifolds with nonsingular Poincaréflows. J Math Anal Appl 145(2):419–430MathSciNetzbMATHGoogle Scholar
  96. 96.
    Hawkins JM (1990) Properties of ergodic flows associated to productodometers. Pac J Math 141:287–294MathSciNetzbMATHGoogle Scholar
  97. 97.
    Hawkins J, Schmidt K (1982) On C 2‑diffeomorphisms of the circle which are of type \( { \mathrm{III}_1 } \). Invent Math66(3):511–518MathSciNetADSzbMATHGoogle Scholar
  98. 98.
    Hawkins J, Silva CE (1997) Characterizing mildly mixing actions by orbit equivalence of products. In: Proceedings of the New York Journal of Mathematics Conference, June 9–13 1997. N Y J Math 3A:99–115MathSciNetGoogle Scholar
  99. 99.
    Hawkins J, Woods EJ (1984) Approximately transitive diffeomorphisms of thecircle. Proc Amer Math Soc 90(2):258–262MathSciNetzbMATHGoogle Scholar
  100. 100.
    Herman M (1979) Construction de difféomorphismes ergodiques, preprintGoogle Scholar
  101. 101.
    Herman M-R (1979) Sur la conjugaison differentiable des diffeomorphismes ducercle a des rotations. Inst Hautes Etudes Sci Publ Math 49:5–233, in FrenchMathSciNetzbMATHGoogle Scholar
  102. 102.
    Herman RH, Putnam IF, Skau CF (1992) Ordered Bratteli diagrams, dimensiongroups and topological dynamics. Int J Math 3(6):827–864MathSciNetzbMATHGoogle Scholar
  103. 103.
    Host B, Méla J-F, Parreau F (1986) Analyse harmonique des mesures. Astérisque 135–136:1–261Google Scholar
  104. 104.
    Host B, Méla J-F, Parreau F (1991) Nonsingular transformations and spectralanalysis of measures. Bull Soc Math France 119:33–90Google Scholar
  105. 105.
    Hurewicz W (1944) Ergodic theorem without invariant measure. Ann Math45:192–206MathSciNetzbMATHGoogle Scholar
  106. 106.
    Inoue K (2004) Isometric extensions and multiple recurrence of infinitemeasure preserving systems. Isr J Math 140:245–252zbMATHGoogle Scholar
  107. 107.
    Ionescu Tulcea A (1965) On the category of certain classes oftransformations in ergodic theory. Trans Amer Math Soc 114:261–279MathSciNetGoogle Scholar
  108. 108.
    Ismagilov RS (1987) Application of a group algebra to problems on thetail σ‑algebra of a random walk on a group and to problems on the ergodicity of a skew action. Izv Akad Nauk SSSR Ser Mat51(4):893–907MathSciNetGoogle Scholar
  109. 109.
    Jaworsky W (1994) Strongly approximatevely transitive actions, theChoquet–Deny theorem, and polynomial growth. Pac J Math 165:115–129Google Scholar
  110. 110.
    James J, Koberda T, Lindsey K, Silva CE, Speh P (2008) Measurable sensitivity. Proc Amer Math Soc 136(10):3549–3559MathSciNetzbMATHGoogle Scholar
  111. 111.
    Janvresse E, Meyerovitch T, de la Rue T, Roy E: Poisson suspensions and entropy of infinite transformations, preprintGoogle Scholar
  112. 112.
    Kaimanovich VA, Vershik AM (1983) Random walks on groups: boundary andentropy. Ann Probab 11:457–490MathSciNetzbMATHGoogle Scholar
  113. 113.
    Kakutani S, Parry W (1963) Infinite measure preserving transformations with“mixing”. Bull Amer Math Soc 69:752–756MathSciNetzbMATHGoogle Scholar
  114. 114.
    Katznelson Y (1977) Sigma‐finite invariant measures for smoothmappings of the circle. J Anal Math 31:1–18MathSciNetzbMATHGoogle Scholar
  115. 115.
    Katznelson Y (1979) The action of diffeomorphism of the circle on theLebesgue measure. J Anal Math 36:156–166MathSciNetzbMATHGoogle Scholar
  116. 116.
    Katznelson Y, Weiss B (1972) The construction of quasi‐invariantmeasures. Isr J Math 12:1–4MathSciNetzbMATHGoogle Scholar
  117. 117.
    Katznelson Y, Weiss B (1991) The classification of nonsingular actions,revisited. Ergod Theory Dynam Syst 11:333–348MathSciNetzbMATHGoogle Scholar
  118. 118.
    Kirillov AA (1978) Elements of the theory of representations. Nauka,MoscowGoogle Scholar
  119. 119.
    Krengel U (1967) Entropy of conservativetransformations. Z Wahrscheinlichkeitstheorie Verw Gebiete 7:161–181Google Scholar
  120. 120.
    Krengel U (1969) Darstellungssätze für Strömungen und Halbströmungen, volII. Math Ann 182:1–39MathSciNetzbMATHGoogle Scholar
  121. 121.
    Krengel U (1970) Transformations without finite invariant measure have finite strong generators. In: Contributions to Ergodic Theory and Probability. Proc Conf Ohio State Univ, Columbus, Ohio. Springer, Berlin, pp 133–157Google Scholar
  122. 122.
    Krengel U (1976) On Rudolph's representation of aperiodic flows. Ann InstH Poincaré Sect B (NS) 12(4):319–338MathSciNetzbMATHGoogle Scholar
  123. 123.
    Krengel U (1985) Ergodic Theorems. De Gruyter Studies in Mathematics,BerlinzbMATHGoogle Scholar
  124. 124.
    Krengel U, Sucheston L (1969) On mixing in infinite measurespaces. Z Wahrscheinlichkeitstheorie Verw Gebiete 13:150–164Google Scholar
  125. 125.
    Krieger W (1969) On nonsingular transformations of a measure space, volI, II. Z Wahrscheinlichkeitstheorie Verw Gebiete 11:83–119Google Scholar
  126. 126.
    Krieger W (1970) On the Araki–Woods asymptotic ratio set and nonsingular transformations of a measure space. In: Contributions to Ergodic Theory and Probability. Proc Conf Ohio State Univ, Columbus, Ohio. In: Lecture Notes in Math, vol 160. Springer, Berlin, pp 158–177Google Scholar
  127. 127.
    Krieger W (1972) On the infinite product construction of nonsingulartransformations of a measure space. Invent Math 15:144–163; Erratum in 26:323–328MathSciNetADSzbMATHGoogle Scholar
  128. 128.
    Krieger W (1976) On Borel automorphisms and their quasi‐invariantmeasures. Math Z 151:19–24MathSciNetzbMATHGoogle Scholar
  129. 129.
    Krieger W (1976) On ergodic flows and isomorphism of factors. Math Ann223:19–70MathSciNetzbMATHGoogle Scholar
  130. 130.
    Kubo I (1969) Quasi-flows. Nagoya Math J35:1–30MathSciNetzbMATHGoogle Scholar
  131. 131.
    Ledrappier F, Sarig O (2007) Invariant measures for the horocycle flow onperiodic hyperbolic surfaces. Isr J Math 160:281–315MathSciNetzbMATHGoogle Scholar
  132. 132.
    Lehrer E, Weiss B (1982) An ε-free Rokhlin lemma. Ergod TheoryDynam Syst 2:45–48MathSciNetzbMATHGoogle Scholar
  133. 133.
    Lemańczyk M, Parreau F (2003) Rokhlin extensions and liftingdisjointness. Ergod Theory Dynam Syst 23:1525–1550Google Scholar
  134. 134.
    Lemańczyk M, Parreau F, Thouvenot J-P (2000) Gaussian automorphismswhose ergodic self‐joinings are Gaussian. Fund Math 164:253–293Google Scholar
  135. 135.
    Mackey GW (1966) Ergodic theory and virtual group. Math Ann166:187–207MathSciNetzbMATHGoogle Scholar
  136. 136.
    Maharam D (1964) Incompressible transformations. Fund MathLVI:35–50MathSciNetGoogle Scholar
  137. 137.
    Mandrekar V, Nadkarni M (1969) On ergodic quasi‐invariant measures onthe circle group. J Funct Anal 3:157–163MathSciNetzbMATHGoogle Scholar
  138. 138.
    Matui H (2002) Topological orbit equivalence of locally compact Cantorminimal systems. Ergod Theory Dynam Syst 22:1871–1903MathSciNetzbMATHGoogle Scholar
  139. 139.
    Méla J-F (1983) Groupes de valeurs propres des systèmes dynamiques etsous‐groupes saturés du cercle. CR Acad Sci Paris Sér I Math 296(10):419–422Google Scholar
  140. 140.
    Meyerovitch T (2007) Extensions and Multiple Recurrence of infinite measure preserving systems, preprint. ArXiv: http://arxiv.org/abs/math/0703914
  141. 141.
    Moore CC (1967) Invariant measures on product spaces. Proc Fifth Berkeley Symp. University of California Press, Berkeley, pp 447–459Google Scholar
  142. 142.
    Moore CC (1982) Ergodic theory and von Neumann algebras. Proc Symp Pure Math38:179–226Google Scholar
  143. 143.
    Moore CC, Schmidt K (1980) Coboundaries and homomorphisms for nonsingularactions and a problem of H Helson. Proc Lond Math Soc 3 40:443–475MathSciNetGoogle Scholar
  144. 144.
    Mortiss G (2000) A non‐singular inverse Vitali lemma withapplications. Ergod Theory Dynam Syst 20:1215–1229MathSciNetzbMATHGoogle Scholar
  145. 145.
    Mortiss G (2002) Average co‐ordinate entropy. J Aust Math Soc73:171–186MathSciNetzbMATHGoogle Scholar
  146. 146.
    Mortiss G (2003) An invariant for nonsingular isomorphism. Ergod TheoryDynam Syst 23:885–893MathSciNetzbMATHGoogle Scholar
  147. 147.
    Nadkarni MG (1979) On spectra of nonsingular transformations andflows. Sankhya Ser A 41(1–2):59–66MathSciNetzbMATHGoogle Scholar
  148. 148.
    Nadkarni MG (1998) Spectral theory of dynamical systems. In: Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, BaselGoogle Scholar
  149. 149.
    Ornstein D (1960) On invariant measures. Bull Amer Math Soc 66:297–300MathSciNetzbMATHGoogle Scholar
  150. 150.
    Ornstein D (1972) On the Root Problem in Ergodic Theory. In: Proc Sixth Berkeley Symp Math Stat Probab. University of California Press, Berkley, pp 347–356Google Scholar
  151. 151.
    Osikawa M (1977/78) Point spectra of nonsingular flows. Publ Res Inst Math Sci13:167–172Google Scholar
  152. 152.
    Osikawa M (1988) Ergodic properties of product type odometers. Springer LectNotes Math 1299:404–414MathSciNetGoogle Scholar
  153. 153.
    Osikawa M, Hamachi T (1971) On zero type and positive type transformationswith infinite invariant measures. Mem Fac Sci Kyushu Univ 25:280–295MathSciNetzbMATHGoogle Scholar
  154. 154.
    Parreau F, Roy E: Poisson joinings of Poisson suspensions, preprintGoogle Scholar
  155. 155.
    Parry W (1963) An ergodic theorem of information theory without invariantmeasure. Proc Lond Math Soc 3 13:605–612MathSciNetGoogle Scholar
  156. 156.
    Parry W (1965) Ergodic and spectral analysis of certain infinite measurepreserving transformations. Proc Amer Math Soc 16:960–966MathSciNetzbMATHGoogle Scholar
  157. 157.
    Parry W (1966) Generators and strong generators in ergodic theory. Bull AmerMath Soc 72:294–296MathSciNetzbMATHGoogle Scholar
  158. 158.
    Parry W (1969) Entropy and generators in ergodic theory. WA Benjamin, NewYork, AmsterdamzbMATHGoogle Scholar
  159. 159.
    Parthasarathy KR, Schmidt K (1977) On the cohomology of a hyperfiniteaction. Monatsh Math 84(1):37–48MathSciNetzbMATHGoogle Scholar
  160. 160.
    Ramsay A (1971) Virtual groups and group actions. Adv Math6:243–322MathSciNetGoogle Scholar
  161. 161.
    Rokhlin VA (1949) Selected topics from the metric theory of dynamicalsystems. Uspekhi Mat Nauk 4:57–125zbMATHGoogle Scholar
  162. 162.
    Rokhlin VA (1965) Generators in ergodic theory, vol II. Vestnik LeningradUniv 20(13):68–72, in Russian, English summaryMathSciNetzbMATHGoogle Scholar
  163. 163.
    Rosinsky J (1995) On the structure of stationary stable processes. AnnProbab 23:1163–1187MathSciNetGoogle Scholar
  164. 164.
    Roy E (2005) Mesures de Poisson, infinie divisibilité et propriétésergodiques. Thèse de doctorat de l'Université Paris 6Google Scholar
  165. 165.
    Roy E (2007) Ergodic properties of Poissonian ID processes. Ann Probab35:551–576MathSciNetzbMATHGoogle Scholar
  166. 166.
    Roy E: Poisson suspensions and infinite ergodic theory, preprintGoogle Scholar
  167. 167.
    Rudolph DJ (1985) Restricted orbit equivalence. Mem Amer Math Soc 54(323)MathSciNetGoogle Scholar
  168. 168.
    Rudolph D, Silva CE (1989) Minimal self‐joinings for nonsingulartransformations. Ergod Theory Dynam Syst 9:759–800MathSciNetzbMATHGoogle Scholar
  169. 169.
    Ryzhikov VV (1994) Factorization of an automorphism of a full Booleanalgebra into the product of three involutions. Mat Zametki 54(2):79–84,159; in Russian. Translation in: Math Notes54(1–2):821–824MathSciNetGoogle Scholar
  170. 170.
    Sachdeva U (1971) On category of mixing in infinite measure spaces. MathSyst Theory 5:319–330MathSciNetzbMATHGoogle Scholar
  171. 171.
    Samorodnitsky G (2005) Null flows, positive flows and the structure ofstationary symmetric stable processes. Ann Probab 33:1782–1803MathSciNetGoogle Scholar
  172. 172.
    Sarig O (2004) Invariant measures for the horocycle flows on Abeliancovers. Invent Math 157:519–551MathSciNetADSzbMATHGoogle Scholar
  173. 173.
    Schmidt K (1977) Cocycles on ergodic transformation groups. MacmillanLectures in Mathematics, vol 1. Macmillan Company of India, DelhiGoogle Scholar
  174. 174.
    Schmidt K (1977) Infinite invariant measures in the circle. Symp Math21:37–43ADSGoogle Scholar
  175. 175.
    Schmidt K (1982) Spectra of ergodic group actions. Isr J Math41(1–2):151–153zbMATHGoogle Scholar
  176. 176.
    Schmidt K (1984) On recurrence. Z Wahrscheinlichkeitstheorie VerwGebiete 68:75–95Google Scholar
  177. 177.
    Schmidt K, Walters P (1982) Mildly mixing actions of locally compactgroups. Proc Lond Math Soc 45:506–518MathSciNetzbMATHGoogle Scholar
  178. 178.
    Shelah S, Weiss B (1982) Measurable recurrence and quasi‐invariantmeasures. Isr Math J 43:154–160MathSciNetzbMATHGoogle Scholar
  179. 179.
    Silva CE, Thieullen P (1991) The subadditive ergodic theorem and recurrence properties of Markovian transformations. J Math Anal Appl 154(1):83–99MathSciNetzbMATHGoogle Scholar
  180. 180.
    Silva CE, Thieullen P (1995) A skew product entropy for nonsingulartransformations. J Lond Math Soc 2 52:497–516MathSciNetGoogle Scholar
  181. 181.
    Silva CE, Witte D (1992) On quotients of nonsingular actions whoseself‐joinings are graphs. Int J Math5:219–237MathSciNetGoogle Scholar
  182. 182.
    Thouvenot J-P (1995) Some properties and applications of joinings in ergodic theory. In: Ergodic theory and its connections with harmonic analysis (Alexandia, 1993), pp 207–235. Lond Math Soc Lect Notes Ser 205. Cambridge Univ Press, CambridgeGoogle Scholar
  183. 183.
    Ullman D (1987) A generalization of a theorem of Atkinson tonon‐invariant measures. Pac J Math 130:187–193MathSciNetzbMATHGoogle Scholar
  184. 184.
    Vershik AM (1983) Manyvalued mappings with invariant measure (polymorphisms)and Markov processes. J Sov Math 23:2243–2266zbMATHGoogle Scholar
  185. 185.
    Vershik AM, Kerov SV (1985) Locally semisimple algebras. In: Combinatorial theory and K 0‑functor. Mod Probl Math 26:3–56MathSciNetGoogle Scholar
  186. 186.
    Zimmer RJ (1977) Random walks on compact groups and the existence ofcocycles. Isr J Math 26:84–90MathSciNetzbMATHGoogle Scholar
  187. 187.
    Zimmer RJ (1978) Amenable ergodic group actions and an application toPoisson boundaries of random walks. J Funct Anal 27:350–372MathSciNetzbMATHGoogle Scholar
  188. 188.
    Zimmer RJ (1984) Ergodic theory and semisimple Lie groups. Birkhäuser, Basel, BostonGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Alexandre I. Danilenko
    • 1
  • Cesar E. Silva
    • 2
  1. 1.Institute for Low Temperature Physics & EngineeringUkrainian National Academy of SciencesKharkovUkraine
  2. 2.Department of MathematicsWilliams CollegeWilliamstownUSA