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Encyclopedia of Complexity and Systems Science pp 3055–3083Cite as

Ergodic Theory : Non-singular Transformations

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Definition of the Subject

An abstract measurable dynamical system consists of a set X (phase space) with a transformation\( { T \colon X \to X } \) (evolution law ortime) and a finite or σ‑finite measure μ on X that specifies a class ofnegligible subsets. Nonsingular ergodic theory studies systems where Trespects μ in a weak sense:the transformation preserves only the class of negligible subsets but it may not preserve μ. This survey is about dynamics and invariants ofnonsingular systems. Such systems model ‘non‐equilibrium’ situations in which events that are impossible at some time remain impossibleat any other time. Of course, the first question that arises is whether it is possible to find an equivalent invariant measure, i. e. pass toa hidden equilibrium without changing the negligible subsets? It turns out that there exist systems which do not admit an equivalent invariant finiteor even σ‑finite measure. They are of our primary interest here. In a way (Baire category) most of...

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Notes

  1. 1.

    This abbreviates ‘infinite tensor product of factors of type I’ (came from the theory of von Neumann algebras).

Abbreviations

Nonsingular dynamical system:

Let \( { (X, \mathcal{B}, \mu) } \) be a standard Borel space equipped with a σ‑finite measure. A Borel map \( { T \colon X \to X } \) is a nonsingular transformation of X if for any \( { N \in \mathcal{B} } \), \( { \mu(T^{-1} N) = 0 } \) if and only if \( { \mu(N) = 0 } \). In this case the measure μ is called quasi‐invariant for T; and the quadruple \( { (X, \mathcal{B}, \mu, T) } \) is called a nonsingular dynamical system. If \( { \mu(A) = \mu(T^{-1}A) } \) for all \( { A \in \mathcal{B} } \) then μ is said to be invariant under T or, equivalently, T is measure‐preserving.

Conservativeness:

T is conservative if for all sets A of positive measure there exists an integer \( { n \mathchar"313E 0 } \) such that \( { \mu(A \cap T^{-n} A) \mathchar"313E 0 } \).

Ergodicity:

T is ergodic if every measurable subset A of X that is invariant under T (i. e., \( { T^{-1}A = A } \)) is either μ-null or μ‑conull. Equivalently, every Borel function \( { f \colon X \to \mathbb{R} } \) such that \( { f \circ T = f } \) is constant a. e.

Types II, \( { \mathrm{II}_1 } \), \( { \mathrm{II}_\infty } \) and III:

Suppose that μ is non‐atomic and T ergodic (and hence conservative). If there exists a σ‑finite measure ν on \( { \mathcal{B} } \) which is equivalent to μ and invariant under T then T is said to be of type II. It is easy to see that ν is unique up to scaling. If ν is finite then T is of type II 1. If ν is infinite then T is of type \( { \mathrm{II}_\infty } \). If T is not of type II then T is said to be of type III.

Bibliography

  1. Aaronson J (1983) The eigenvalues of nonsingular transformations. IsrJ Math 45:297–312

    MathSciNet  MATH  Google Scholar 

  2. Aaronson J (1987) The intrinsic normalizing constants of transformationspreserving infinite measures. J Analyse Math 49:239–270

    MathSciNet  MATH  Google Scholar 

  3. Aaronson J (1997) An Introduction to Infinite Ergodic Theory. Amer Math Soc,Providence

    MATH  Google Scholar 

  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 77–88

    Google Scholar 

  5. Aaronson J, Lin M, Weiss B (1979) Mixing properties of Markov operators andergodic transformations, and ergodicity of cartesian products. Isr J Math 33:198–224

    MathSciNet  MATH  Google Scholar 

  6. Aaronson J, Nadkarni M (1987) \( { L_\infty } \) eigenvalues and L 2spectra of nonsingular transformations. Proc Lond Math Soc 55(3):538–570

    MathSciNet  MATH  Google Scholar 

  7. Aaronson J, Nakada H (2000) Multiple recurrence of Markov shifts and otherinfinite measure preserving transformations. Isr J Math 117:285–310

    MathSciNet  MATH  Google Scholar 

  8. Aaronson J, Park KK (2008) Predictability, entropy and information of infinite transformations, preprint. ArXiv:0705:2148v3

    Google Scholar 

  9. Aaronson J, Weiss B (2004) On Herman's theorem for ergodic, amenable groupextensions of endomorphisms. Ergod Theory Dynam Syst 5:1283–1293

    MathSciNet  Google Scholar 

  10. Adams S, Elliott GA, Giordano T (1994) Amenable actions of groups. Trans AmerMath Soc 344:803–822

    MathSciNet  MATH  Google Scholar 

  11. Adams T, Friedman N, Silva CE (1997) Rank-One Weak Mixing for NonsingularTransformations. Isr J Math 102:269–281

    MathSciNet  MATH  Google Scholar 

  12. Adams T, Friedman N, Silva CE (2001) Rank one power weak mixing fornonsingular transformations. Ergod Theory Dynam Systems 21:1321–1332

    MathSciNet  MATH  Google Scholar 

  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):357–365

    MathSciNet  Google Scholar 

  14. Albeverio S, Hoegh-Krohn R, Testard D, Vershik AM (1983) Factorialrepresentations of Path groups. J Funct Anal 51:115–231

    MathSciNet  MATH  Google Scholar 

  15. Atkinson G (1976) Recurrence of co‐cycles and random walks. J LondMath Soc 13:486–488

    MATH  Google Scholar 

  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 1–32

    Google Scholar 

  17. Bergelson V, Leibman A (1996) Polynomial extensions of van der Waerden's andSemerédi's theorems. J Amer Math Soc 9:725–753

    MathSciNet  MATH  Google Scholar 

  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):341–369

    MathSciNet  MATH  Google Scholar 

  19. Bezuglyi SI, Golodets VY (1991) Weak equivalence and the structures ofcocycles of an ergodic automorphism. Publ Res Inst Math Sci 27(4):577–625

    MathSciNet  Google Scholar 

  20. Bowles A, Fidkowski L, Marinello A, Silva CE (2001) Double ergodicity ofnonsingular transformations and infinite measure‐preserving staircase transformations. Ill J Math45(3):999–1019

    MathSciNet  MATH  Google Scholar 

  21. Chacon RV, Friedman NA (1965) Approximation and invariant measures. ZWahrscheinlichkeitstheorie Verw Gebiete 3:286–295

    Google Scholar 

  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 73–85

    Google Scholar 

  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 103:187–205

    MathSciNet  MATH  Google Scholar 

  24. Choksi JR, Kakutani S (1979) Residuality of ergodic measurable transformationsand of ergodic transformations which preserve an infinite measure. Ind Univ Math J 28:453–469

    MathSciNet  MATH  Google Scholar 

  25. Choksi JR, Nadkarni MG (1994) The maximal spectral type of a rank onetransformation. Canad Math Bull 37(1):29–36

    MathSciNet  MATH  Google Scholar 

  26. Choksi JR, Nadkarni MG (2000) Genericity of nonsingular transformations withinfinite ergodic index. Colloq Math 84/85:195–201

    MathSciNet  Google Scholar 

  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 94–113

    Google Scholar 

  28. Connes A (1975) On the hierarchy of W Krieger. Ill J Math19:428–432

    MathSciNet  MATH  Google Scholar 

  29. Connes A, Feldman J, Weiss B (1981) An amenable equivalence relation isgenerated by a single transformation. Ergod Theory Dynam Systems 1:431–450

    MathSciNet  MATH  Google Scholar 

  30. Connes A, Krieger W (1977) Measure space automorphisms, the normalizers oftheir full groups, and approximate finiteness. J Funct Anal 24(4):336–352

    MathSciNet  MATH  Google Scholar 

  31. Connes A, Woods EJ (1985) Approximately transitive flows and ITPFIfactors. Ergod Theory Dynam Syst 5(2):203–236

    MathSciNet  MATH  Google Scholar 

  32. Connes A, Woods EJ (1989) Hyperfinite von Neumann algebras and Poissonboundaries of time dependent random walks. Pac J Math 37:225–243

    MathSciNet  Google Scholar 

  33. Cornfeld IP, Fomin VS, Sinaĭ YG (1982) Ergodic theory. Grundlehren der Mathematischen Wissenschaften, vol 245. Springer, New York

    Google Scholar 

  34. Danilenko AI (1995) The topological structure of Polish groups and groupoidsof measure space transformations. Publ Res Inst Math Sci 31(5):913–940

    MathSciNet  MATH  Google Scholar 

  35. Danilenko AI (1998) Quasinormal subrelations of ergodic equivalencerelations. Proc Amer Math Soc 126(11):3361–3370

    MathSciNet  MATH  Google Scholar 

  36. Danilenko AI (2001) Funny rank one weak mixing for nonsingular Abelianactions. Isr J Math 121:29–54

    MathSciNet  MATH  Google Scholar 

  37. Danilenko AI (2001) Strong orbit equivalence of locally compact Cantor minimalsystems. Int J Math 12:113–123

    MathSciNet  MATH  Google Scholar 

  38. Danilenko AI (2004) Infinite rank one actions and nonsingular Chacontransformations. Ill J Math 48(3):769–786

    MathSciNet  MATH  Google Scholar 

  39. Danilenko AI (2007) On simplicity concepts for ergodic actions. J d'AnalMath 102:77–118

    Google Scholar 

  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 265:325–351

    MathSciNet  Google Scholar 

  41. Danilenko AI, Golodets YV (1996) On extension of cocycles to normalizerelements, outer conjugacy, and related problems. Trans Amer Math Soc 348(12):4857–4882

    MathSciNet  MATH  Google Scholar 

  42. Danilenko AI, Hamachi T (2000) On measure theoretical analogues of theTakesaki structure theorem for type III factors. Colloq Math 84/85:485–493

    MathSciNet  Google Scholar 

  43. Danilenko AI, Lemańczyk M (2005) A class of multipliers for\( { \mathcal{W}^\perp } \). Isr J Math148:137–168

    Google Scholar 

  44. Danilenko AI, Rudolph DJ: Conditional entropy theory in infinite measure and a question of Krengel. Isr J Math, to appear

    Google Scholar 

  45. Danilenko AI, Silva CE (2004) Multiple and polynomial recurrence for Abelianactions in infinite measure. J Lond Math Soc 2 69(1):183–200

    MathSciNet  Google Scholar 

  46. Danilenko AI, Solomko AV: Infinite measure preserving flows with infinite ergodic index. Colloq Math, to appear

    Google Scholar 

  47. Day S, Grivna B, McCartney E, Silva CE (1999) Power Weakly Mixing InfiniteTransformations. N Y J Math 5:17–24

    MathSciNet  MATH  Google Scholar 

  48. del Junco A (1978) A simple measure‐preserving transformationwith trivial centralizer. Pac J Math 79:357–362

    Google Scholar 

  49. del Junco A, Rudolph DJ (1987) On ergodic actions whose self‐joiningsare graphs. Ergod Theory Dynam Syst 7:531–557

    MATH  Google Scholar 

  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 101–115

    Google Scholar 

  51. del Junco A, Silva CE (2003) On factors of nonsingular Cartesianproducts. Ergod Theory Dynam Syst 23(5):1445–1465

    MATH  Google Scholar 

  52. Derriennic Y, Frączek K, Lemańczyk M, Parreau F (2008) Ergodicautomorphisms whose weak closure of off‐diagonal measures consists of ergodic self‐joinings. Colloq Math110:81–115

    Google Scholar 

  53. Dixmier J (1969) Les \( { C^* } \)-algèbres et leurs représentations. Gauthier‐Villars Editeur,Paris

    Google Scholar 

  54. Dooley AH, Hamachi T (2003) Nonsingular dynamical systems, Bratteli diagramsand Markov odometers. Isr J Math 138:93–123

    MathSciNet  MATH  Google Scholar 

  55. Dooley AH, Hamachi T (2003) Markov odometer actions not of product type. ErgodTheory Dynam Syst 23(3):813–829

    MathSciNet  MATH  Google Scholar 

  56. Dooley AH, Klemes I, Quas AN (1998) Product and Markov measures oftype III. J Aust Math Soc Ser A 65(1):84–110

    MathSciNet  MATH  Google Scholar 

  57. Dooley AH, Mortiss G: On the critical dimension of product odometers, preprint

    Google Scholar 

  58. Dooley AH, Mortiss G (2006) On the critical dimension and AC entropy forMarkov odometers. Monatsh Math 149:193–213

    MathSciNet  MATH  Google Scholar 

  59. Dooley AH, Mortiss G (2007) The critical dimensions of Hamachi shifts. TohokuMath J 59(2):57–66

    MathSciNet  MATH  Google Scholar 

  60. Dye H (1963) On groups of measure‐preserving transformations I. AmerJ Math 81:119–159, and II, Amer J Math 85:551–576

    MathSciNet  Google Scholar 

  61. Effros EG (1965) Transformation groups and \( { C^* } \)-algebras. Ann Math81(2):38–55

    MathSciNet  MATH  Google Scholar 

  62. Eigen SJ (1981) On the simplicity of the full group of ergodictransformations. Isr J Math 40(3–4):345–349

    MathSciNet  MATH  Google Scholar 

  63. Eigen SJ (1982) The group of measure preserving transformations of [0,1] hasno outer automorphisms. Math Ann 259:259–270

    MathSciNet  MATH  Google Scholar 

  64. Eigen S, Hajian A, Halverson K (1998) Multiple recurrence and infinite measurepreserving odometers. Isr J Math 108:37–44

    MathSciNet  MATH  Google Scholar 

  65. Fedorov A (1985) Krieger's theorem for cocycles, preprint

    Google Scholar 

  66. Feldman J, Moore CC (1977) Ergodic equivalence relations, cohomology, and vonNeumann algebras. I. Trans Amer Math Soc 234:289–324

    MathSciNet  MATH  Google Scholar 

  67. Ferenczi S (1985) Systèmes de rang un gauche. Ann Inst H Poincaré ProbabStatist 21(2):177–186

    MathSciNet  MATH  Google Scholar 

  68. Friedman NA (1970) Introduction to Ergodic Theory. Van Nostrand Reinhold Mathematical Studies, No 29. Van Nostrand Reinhold Co., New York

    Google Scholar 

  69. Furstenberg H (1967) Disjointness in ergodic theory, minimal sets anddiophantine approximation. Math Syst Theory 1:1–49

    MathSciNet  MATH  Google Scholar 

  70. Furstenberg H (1981) Recurrence in Ergodic Theory and Combinatorial NumberTheory. Princeton University Press, Princeton

    Google Scholar 

  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 127–132

    Google Scholar 

  72. Gårding L, Wightman AS (1954) Representation of anticommutationrelations. Proc Nat Acad Sci USA 40:617–621

    Google Scholar 

  73. Giordano T, Skandalis G (1985) Krieger factors isomorphic to their tensorsquare and pure point spectrum flows. J Funct Anal 64(2):209–226

    MathSciNet  MATH  Google Scholar 

  74. Giordano T, Skandalis G (1985) On infinite tensor products of factors of typeI 2. Ergod Theory Dynam Syst 5:565–586

    MathSciNet  MATH  Google Scholar 

  75. Glasner E (1994) On the multipliers of \( { \mathcal{W}^\perp } \). Ergod Theory Dynam Syst14:129–140

    MathSciNet  MATH  Google Scholar 

  76. Glimm J (1961) Locally compact transformation groups. Trans Amer Math Soc101:124–138

    MathSciNet  MATH  Google Scholar 

  77. Golodets YV (1969) A description of the representations ofanticommutation relations. Uspehi Matemat Nauk 24(4):43–64

    Google Scholar 

  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 19–83

    Google Scholar 

  79. Golodets YV, Sinel'shchikov SD (1985) Locally compact groups appearing asranges of cocycles of ergodic Z‑actions. Ergod Theory Dynam Syst5:47–57

    MathSciNet  MATH  Google Scholar 

  80. Golodets YV, Sinel'shchikov SD (1990) Amenable ergodic actions of groups andimages of cocycles. Dokl Akad Nauk SSSR 312(6):1296–1299, in Russian

    MathSciNet  Google Scholar 

  81. Golodets YV, Sinel'shchikov SD (1994) Classification and structure of cocyclesof amenable ergodic equivalence relations. J Funct Anal 121(2):455–485

    MathSciNet  MATH  Google Scholar 

  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 9:1–22

    MathSciNet  MATH  Google Scholar 

  83. Hajian AB, Kakutani S (1964) Weakly wandering sets and invariantmeasures. Trans Amer Math Soc 110:136–151

    MathSciNet  MATH  Google Scholar 

  84. Halmos PR (1946) An ergodic theorem. Proc Nat Acad Sci USA32:156–161

    MathSciNet  ADS  MATH  Google Scholar 

  85. Halmos PR (1956) Lectures on ergodic theory. Publ Math Soc Jpn 3

    Google Scholar 

  86. Hamachi T (1981) The normalizer group of an ergodic automorphism oftype III and the commutant of an ergodic flow. J Funct Anal 40:387–403

    MathSciNet  MATH  Google Scholar 

  87. Hamachi T (1981) On a Bernoulli shift with nonidentical factor measures. ErgodTheory Dynam Syst 1:273–283

    MathSciNet  MATH  Google Scholar 

  88. Hamachi T (1992) A measure theoretical proof of the Connes–Woodstheorem on AT-flows. Pac J Math 154:67–85

    MathSciNet  MATH  Google Scholar 

  89. Hamachi T, Kosaki H (1993) Orbital factor map. Ergod Theory Dynam Syst13:515–532

    MathSciNet  MATH  Google Scholar 

  90. Hamachi T, Osikawa M (1981) Ergodic groups of automorphisms and Krieger'stheorems. Semin Math Sci 3, Keio Univ

    Google Scholar 

  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 196–210

    Google Scholar 

  92. Hamachi T, Silva CE (2000) On nonsingular Chacon transformations. IllJ Math 44:868–883

    MathSciNet  MATH  Google Scholar 

  93. Hawkins JM (1982) Non-ITPFI diffeomorphisms. Isr J Math42:117–131

    MathSciNet  MATH  Google Scholar 

  94. Hawkins JM (1983) Smooth type III diffeomorphisms of manifolds. TransAmer Math Soc 276:625–643

    MathSciNet  MATH  Google Scholar 

  95. Hawkins JM (1990) Diffeomorphisms of manifolds with nonsingular Poincaréflows. J Math Anal Appl 145(2):419–430

    MathSciNet  MATH  Google Scholar 

  96. Hawkins JM (1990) Properties of ergodic flows associated to productodometers. Pac J Math 141:287–294

    MathSciNet  MATH  Google Scholar 

  97. Hawkins J, Schmidt K (1982) On C 2‑diffeomorphisms of the circle which are of type \( { \mathrm{III}_1 } \). Invent Math66(3):511–518

    MathSciNet  ADS  MATH  Google Scholar 

  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:99–115

    MathSciNet  Google Scholar 

  99. Hawkins J, Woods EJ (1984) Approximately transitive diffeomorphisms of thecircle. Proc Amer Math Soc 90(2):258–262

    MathSciNet  MATH  Google Scholar 

  100. Herman M (1979) Construction de difféomorphismes ergodiques, preprint

    Google Scholar 

  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 French

    MathSciNet  MATH  Google Scholar 

  102. Herman RH, Putnam IF, Skau CF (1992) Ordered Bratteli diagrams, dimensiongroups and topological dynamics. Int J Math 3(6):827–864

    MathSciNet  MATH  Google Scholar 

  103. Host B, Méla J-F, Parreau F (1986) Analyse harmonique des mesures. Astérisque 135–136:1–261

    Google Scholar 

  104. Host B, Méla J-F, Parreau F (1991) Nonsingular transformations and spectralanalysis of measures. Bull Soc Math France 119:33–90

    Google Scholar 

  105. Hurewicz W (1944) Ergodic theorem without invariant measure. Ann Math45:192–206

    MathSciNet  MATH  Google Scholar 

  106. Inoue K (2004) Isometric extensions and multiple recurrence of infinitemeasure preserving systems. Isr J Math 140:245–252

    MATH  Google Scholar 

  107. Ionescu Tulcea A (1965) On the category of certain classes oftransformations in ergodic theory. Trans Amer Math Soc 114:261–279

    MathSciNet  Google Scholar 

  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):893–907

    MathSciNet  Google Scholar 

  109. Jaworsky W (1994) Strongly approximatevely transitive actions, theChoquet–Deny theorem, and polynomial growth. Pac J Math 165:115–129

    Google Scholar 

  110. James J, Koberda T, Lindsey K, Silva CE, Speh P (2008) Measurable sensitivity. Proc Amer Math Soc 136(10):3549–3559

    MathSciNet  MATH  Google Scholar 

  111. Janvresse E, Meyerovitch T, de la Rue T, Roy E: Poisson suspensions and entropy of infinite transformations, preprint

    Google Scholar 

  112. Kaimanovich VA, Vershik AM (1983) Random walks on groups: boundary andentropy. Ann Probab 11:457–490

    MathSciNet  MATH  Google Scholar 

  113. Kakutani S, Parry W (1963) Infinite measure preserving transformations with“mixing”. Bull Amer Math Soc 69:752–756

    MathSciNet  MATH  Google Scholar 

  114. Katznelson Y (1977) Sigma‐finite invariant measures for smoothmappings of the circle. J Anal Math 31:1–18

    MathSciNet  MATH  Google Scholar 

  115. Katznelson Y (1979) The action of diffeomorphism of the circle on theLebesgue measure. J Anal Math 36:156–166

    MathSciNet  MATH  Google Scholar 

  116. Katznelson Y, Weiss B (1972) The construction of quasi‐invariantmeasures. Isr J Math 12:1–4

    MathSciNet  MATH  Google Scholar 

  117. Katznelson Y, Weiss B (1991) The classification of nonsingular actions,revisited. Ergod Theory Dynam Syst 11:333–348

    MathSciNet  MATH  Google Scholar 

  118. Kirillov AA (1978) Elements of the theory of representations. Nauka,Moscow

    Google Scholar 

  119. Krengel U (1967) Entropy of conservativetransformations. Z Wahrscheinlichkeitstheorie Verw Gebiete 7:161–181

    Google Scholar 

  120. Krengel U (1969) Darstellungssätze für Strömungen und Halbströmungen, volII. Math Ann 182:1–39

    MathSciNet  MATH  Google Scholar 

  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 133–157

    Google Scholar 

  122. Krengel U (1976) On Rudolph's representation of aperiodic flows. Ann InstH Poincaré Sect B (NS) 12(4):319–338

    MathSciNet  MATH  Google Scholar 

  123. Krengel U (1985) Ergodic Theorems. De Gruyter Studies in Mathematics,Berlin

    MATH  Google Scholar 

  124. Krengel U, Sucheston L (1969) On mixing in infinite measurespaces. Z Wahrscheinlichkeitstheorie Verw Gebiete 13:150–164

    Google Scholar 

  125. Krieger W (1969) On nonsingular transformations of a measure space, volI, II. Z Wahrscheinlichkeitstheorie Verw Gebiete 11:83–119

    Google Scholar 

  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 158–177

    Google Scholar 

  127. Krieger W (1972) On the infinite product construction of nonsingulartransformations of a measure space. Invent Math 15:144–163; Erratum in 26:323–328

    MathSciNet  ADS  MATH  Google Scholar 

  128. Krieger W (1976) On Borel automorphisms and their quasi‐invariantmeasures. Math Z 151:19–24

    MathSciNet  MATH  Google Scholar 

  129. Krieger W (1976) On ergodic flows and isomorphism of factors. Math Ann223:19–70

    MathSciNet  MATH  Google Scholar 

  130. Kubo I (1969) Quasi-flows. Nagoya Math J35:1–30

    MathSciNet  MATH  Google Scholar 

  131. Ledrappier F, Sarig O (2007) Invariant measures for the horocycle flow onperiodic hyperbolic surfaces. Isr J Math 160:281–315

    MathSciNet  MATH  Google Scholar 

  132. Lehrer E, Weiss B (1982) An ε-free Rokhlin lemma. Ergod TheoryDynam Syst 2:45–48

    MathSciNet  MATH  Google Scholar 

  133. Lemańczyk M, Parreau F (2003) Rokhlin extensions and liftingdisjointness. Ergod Theory Dynam Syst 23:1525–1550

    Google Scholar 

  134. Lemańczyk M, Parreau F, Thouvenot J-P (2000) Gaussian automorphismswhose ergodic self‐joinings are Gaussian. Fund Math 164:253–293

    Google Scholar 

  135. Mackey GW (1966) Ergodic theory and virtual group. Math Ann166:187–207

    MathSciNet  MATH  Google Scholar 

  136. Maharam D (1964) Incompressible transformations. Fund MathLVI:35–50

    MathSciNet  Google Scholar 

  137. Mandrekar V, Nadkarni M (1969) On ergodic quasi‐invariant measures onthe circle group. J Funct Anal 3:157–163

    MathSciNet  MATH  Google Scholar 

  138. Matui H (2002) Topological orbit equivalence of locally compact Cantorminimal systems. Ergod Theory Dynam Syst 22:1871–1903

    MathSciNet  MATH  Google Scholar 

  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):419–422

    Google Scholar 

  140. Meyerovitch T (2007) Extensions and Multiple Recurrence of infinite measure preserving systems, preprint. ArXiv: http://arxiv.org/abs/math/0703914

  141. Moore CC (1967) Invariant measures on product spaces. Proc Fifth Berkeley Symp. University of California Press, Berkeley, pp 447–459

    Google Scholar 

  142. Moore CC (1982) Ergodic theory and von Neumann algebras. Proc Symp Pure Math38:179–226

    Google Scholar 

  143. Moore CC, Schmidt K (1980) Coboundaries and homomorphisms for nonsingularactions and a problem of H Helson. Proc Lond Math Soc 3 40:443–475

    MathSciNet  Google Scholar 

  144. Mortiss G (2000) A non‐singular inverse Vitali lemma withapplications. Ergod Theory Dynam Syst 20:1215–1229

    MathSciNet  MATH  Google Scholar 

  145. Mortiss G (2002) Average co‐ordinate entropy. J Aust Math Soc73:171–186

    MathSciNet  MATH  Google Scholar 

  146. Mortiss G (2003) An invariant for nonsingular isomorphism. Ergod TheoryDynam Syst 23:885–893

    MathSciNet  MATH  Google Scholar 

  147. Nadkarni MG (1979) On spectra of nonsingular transformations andflows. Sankhya Ser A 41(1–2):59–66

    MathSciNet  MATH  Google Scholar 

  148. Nadkarni MG (1998) Spectral theory of dynamical systems. In: Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, Basel

    Google Scholar 

  149. Ornstein D (1960) On invariant measures. Bull Amer Math Soc 66:297–300

    MathSciNet  MATH  Google Scholar 

  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 347–356

    Google Scholar 

  151. Osikawa M (1977/78) Point spectra of nonsingular flows. Publ Res Inst Math Sci13:167–172

    Google Scholar 

  152. Osikawa M (1988) Ergodic properties of product type odometers. Springer LectNotes Math 1299:404–414

    MathSciNet  Google Scholar 

  153. Osikawa M, Hamachi T (1971) On zero type and positive type transformationswith infinite invariant measures. Mem Fac Sci Kyushu Univ 25:280–295

    MathSciNet  MATH  Google Scholar 

  154. Parreau F, Roy E: Poisson joinings of Poisson suspensions, preprint

    Google Scholar 

  155. Parry W (1963) An ergodic theorem of information theory without invariantmeasure. Proc Lond Math Soc 3 13:605–612

    MathSciNet  Google Scholar 

  156. Parry W (1965) Ergodic and spectral analysis of certain infinite measurepreserving transformations. Proc Amer Math Soc 16:960–966

    MathSciNet  MATH  Google Scholar 

  157. Parry W (1966) Generators and strong generators in ergodic theory. Bull AmerMath Soc 72:294–296

    MathSciNet  MATH  Google Scholar 

  158. Parry W (1969) Entropy and generators in ergodic theory. WA Benjamin, NewYork, Amsterdam

    MATH  Google Scholar 

  159. Parthasarathy KR, Schmidt K (1977) On the cohomology of a hyperfiniteaction. Monatsh Math 84(1):37–48

    MathSciNet  MATH  Google Scholar 

  160. Ramsay A (1971) Virtual groups and group actions. Adv Math6:243–322

    MathSciNet  Google Scholar 

  161. Rokhlin VA (1949) Selected topics from the metric theory of dynamicalsystems. Uspekhi Mat Nauk 4:57–125

    MATH  Google Scholar 

  162. Rokhlin VA (1965) Generators in ergodic theory, vol II. Vestnik LeningradUniv 20(13):68–72, in Russian, English summary

    MathSciNet  MATH  Google Scholar 

  163. Rosinsky J (1995) On the structure of stationary stable processes. AnnProbab 23:1163–1187

    MathSciNet  Google Scholar 

  164. Roy E (2005) Mesures de Poisson, infinie divisibilité et propriétésergodiques. Thèse de doctorat de l'Université Paris 6

    Google Scholar 

  165. Roy E (2007) Ergodic properties of Poissonian ID processes. Ann Probab35:551–576

    MathSciNet  MATH  Google Scholar 

  166. Roy E: Poisson suspensions and infinite ergodic theory, preprint

    Google Scholar 

  167. Rudolph DJ (1985) Restricted orbit equivalence. Mem Amer Math Soc 54(323)

    MathSciNet  Google Scholar 

  168. Rudolph D, Silva CE (1989) Minimal self‐joinings for nonsingulartransformations. Ergod Theory Dynam Syst 9:759–800

    MathSciNet  MATH  Google Scholar 

  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):821–824

    MathSciNet  Google Scholar 

  170. Sachdeva U (1971) On category of mixing in infinite measure spaces. MathSyst Theory 5:319–330

    MathSciNet  MATH  Google Scholar 

  171. Samorodnitsky G (2005) Null flows, positive flows and the structure ofstationary symmetric stable processes. Ann Probab 33:1782–1803

    MathSciNet  Google Scholar 

  172. Sarig O (2004) Invariant measures for the horocycle flows on Abeliancovers. Invent Math 157:519–551

    MathSciNet  ADS  MATH  Google Scholar 

  173. Schmidt K (1977) Cocycles on ergodic transformation groups. MacmillanLectures in Mathematics, vol 1. Macmillan Company of India, Delhi

    Google Scholar 

  174. Schmidt K (1977) Infinite invariant measures in the circle. Symp Math21:37–43

    ADS  Google Scholar 

  175. Schmidt K (1982) Spectra of ergodic group actions. Isr J Math41(1–2):151–153

    MATH  Google Scholar 

  176. Schmidt K (1984) On recurrence. Z Wahrscheinlichkeitstheorie VerwGebiete 68:75–95

    Google Scholar 

  177. Schmidt K, Walters P (1982) Mildly mixing actions of locally compactgroups. Proc Lond Math Soc 45:506–518

    MathSciNet  MATH  Google Scholar 

  178. Shelah S, Weiss B (1982) Measurable recurrence and quasi‐invariantmeasures. Isr Math J 43:154–160

    MathSciNet  MATH  Google Scholar 

  179. Silva CE, Thieullen P (1991) The subadditive ergodic theorem and recurrence properties of Markovian transformations. J Math Anal Appl 154(1):83–99

    MathSciNet  MATH  Google Scholar 

  180. Silva CE, Thieullen P (1995) A skew product entropy for nonsingulartransformations. J Lond Math Soc 2 52:497–516

    MathSciNet  Google Scholar 

  181. Silva CE, Witte D (1992) On quotients of nonsingular actions whoseself‐joinings are graphs. Int J Math5:219–237

    MathSciNet  Google Scholar 

  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, Cambridge

    Google Scholar 

  183. Ullman D (1987) A generalization of a theorem of Atkinson tonon‐invariant measures. Pac J Math 130:187–193

    MathSciNet  MATH  Google Scholar 

  184. Vershik AM (1983) Manyvalued mappings with invariant measure (polymorphisms)and Markov processes. J Sov Math 23:2243–2266

    MATH  Google Scholar 

  185. Vershik AM, Kerov SV (1985) Locally semisimple algebras. In: Combinatorial theory and K 0‑functor. Mod Probl Math 26:3–56

    MathSciNet  Google Scholar 

  186. Zimmer RJ (1977) Random walks on compact groups and the existence ofcocycles. Isr J Math 26:84–90

    MathSciNet  MATH  Google Scholar 

  187. Zimmer RJ (1978) Amenable ergodic group actions and an application toPoisson boundaries of random walks. J Funct Anal 27:350–372

    MathSciNet  MATH  Google Scholar 

  188. Zimmer RJ (1984) Ergodic theory and semisimple Lie groups. Birkhäuser, Basel, Boston

    Google Scholar 

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Authors and Affiliations

  1. Institute for Low Temperature Physics & Engineering, Ukrainian National Academy of Sciences, Kharkov, Ukraine

    Alexandre I. Danilenko

  2. Department of Mathematics, Williams College, Williamstown, USA

    Cesar E. Silva

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  1. Alexandre I. Danilenko
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Danilenko, A.I., Silva, C.E. (2009). Ergodic Theory : Non-singular Transformations . In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_183

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