Skip to main content
SpringerLink
Log in

Ergodic Theory : Non-singular Transformations

  • Reference work entry
Mathematics of Complexity and Dynamical Systems

Article Outline

Glossary

Definition of the Subject

Basic Results

Panorama of Examples

Mixing Notions and multiple recurrence

Topological Group Aut(X, μ)

Orbit Theory

Smooth Nonsingular Transformations

Spectral Theory for Nonsingular Systems

Entropy and Other Invariants

Nonsingular Joinings and Factors

Applications. Connections with Other Fields

Concluding Remarks

Bibliography

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 600.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

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. Isr J Math 45:297–312

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  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 and ergodic transformations, and ergodicity of cartesian products. Isr J Math 33:198–224

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  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 group extensions of endomorphisms. Ergod Theory Dynam Syst 5:1283–1293

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Ageev ON, Silva CE (2002) Genericity of rigid and multiply recurrent infinite measure‐preserving and nonsingular transformations. In: Proceedings of the 16th Summer Conference on General Topology and its Applications. Topology Proc 26(2):357–365

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  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 and Semerédi's theorems. J Amer Math Soc 9:725–753

    Article  MathSciNet  MATH  Google Scholar 

  18. Bezuglyi SI, Golodets VY (1985) Groups of measure space transformations and invariants of outer conjugation for automorphisms from normalizers of type III full groups. J Funct Anal 60(3):341–369

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Bowles A, Fidkowski L, Marinello A, Silva CE (2001) Double ergodicity of nonsingular transformations and infinite measure‐preserving staircase transformations. Ill J Math 45(3):999–1019

    MathSciNet  MATH  Google Scholar 

  21. Chacon RV, Friedman NA (1965) Approximation and invariant measures. Z Wahrscheinlichkeitstheorie 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

    Chapter  Google Scholar 

  23. Choksi JR, Hawkins JM, Prasad VS (1987) Abelian cocylces for nonsingular ergodic transformations and the genericity of type \({\mathrm{III}_1}\) transformations. Monat fur Math 103:187–205

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Choksi JR, Nadkarni MG (2000) Genericity of nonsingular transformations with infinite 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

    Chapter  Google Scholar 

  28. Connes A (1975) On the hierarchy of W Krieger. Ill J Math 19:428–432

    MathSciNet  MATH  Google Scholar 

  29. Connes A, Feldman J, Weiss B (1981) An amenable equivalence relation is generated 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 of their full groups, and approximate finiteness. J Funct Anal 24(4):336–352

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  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 groupoids of measure space transformations. Publ Res Inst Math Sci 31(5):913–940

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  39. Danilenko AI (2007) On simplicity concepts for ergodic actions. J d'Anal Math 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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Danilenko AI, Hamachi T (2000) On measure theoretical analogues of the Takesaki 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 Math 148:137–168

    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 Abelian actions in infinite measure. J Lond Math Soc 2 69(1):183–200

    Article  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 Infinite Transformations. N Y J Math 5:17–24

    MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  49. del Junco A, Rudolph DJ (1987) On ergodic actions whose self‐joinings are 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 of coding techniques applied to nonsingular maps. In: Takahashi Y (ed) Fractals and Dynamics. Plenum, New York, pp 101–115

    Chapter  Google Scholar 

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

    Article  MATH  Google Scholar 

  52. Derriennic Y, Frączek K, Lemańczyk M, Parreau F (2008) Ergodic automorphisms whose weak closure of off‐diagonal measures consists of ergodic self‐joinings. Colloq Math 110:81–115

    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 diagrams and Markov odometers. Isr J Math 138:93–123

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  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 for Markov odometers. Monatsh Math 149:193–213

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  61. Effros EG (1965) Transformation groups and \({C^*}\)-algebras. Ann Math 81(2):38–55

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  63. Eigen SJ (1982) The group of measure preserving transformations of [0,1] has no outer automorphisms. Math Ann 259:259–270

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  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 von Neumann algebras. I. Trans Amer Math Soc 234:289–324

    Article  MathSciNet  MATH  Google Scholar 

  67. Ferenczi S (1985) Systèmes de rang un gauche. Ann Inst H Poincaré Probab Statist 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 and diophantine approximation. Math Syst Theory 1:1–49

    Article  MathSciNet  MATH  Google Scholar 

  70. Furstenberg H (1981) Recurrence in Ergodic Theory and Combinatorial Number Theory. 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 anticommutation relations. Proc Nat Acad Sci USA 40:617–621

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  75. Glasner E (1994) On the multipliers of \({\mathcal{W}^\perp}\). Ergod Theory Dynam Syst 14:129–140

    Article  MathSciNet  MATH  Google Scholar 

  76. Glimm J (1961) Locally compact transformation groups. Trans Amer Math Soc 101:124–138

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  78. Golodets YV, Sinel'shchikov SD (1983) Existence and uniqueness of cocycles of ergodic 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 as ranges of cocycles of ergodic Z‑actions. Ergod Theory Dynam Syst 5:47–57

    Article  MathSciNet  MATH  Google Scholar 

  80. Golodets YV, Sinel'shchikov SD (1990) Amenable ergodic actions of groups and images 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 cocycles of amenable ergodic equivalence relations. J Funct Anal 121(2):455–485

    Article  MathSciNet  MATH  Google Scholar 

  82. Gruher K, Hines F, Patel D, Silva CE, Waelder R (2003) Power weak mixing does not 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 invariant measures. Trans Amer Math Soc 110:136–151

    Article  MathSciNet  MATH  Google Scholar 

  84. Halmos PR (1946) An ergodic theorem. Proc Nat Acad Sci USA 32:156–161

    Article  MathSciNet  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 of type III and the commutant of an ergodic flow. J Funct Anal 40:387–403

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  89. Hamachi T, Kosaki H (1993) Orbital factor map. Ergod Theory Dynam Syst 13:515–532

    Article  MathSciNet  MATH  Google Scholar 

  90. Hamachi T, Osikawa M (1981) Ergodic groups of automorphisms and Krieger's theorems. 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. Ill J Math 44:868–883

    MathSciNet  MATH  Google Scholar 

  93. Hawkins JM (1982) Non-ITPFI diffeomorphisms. Isr J Math 42:117–131

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  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 Math 66(3):511–518

    Article  MathSciNet  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 the circle. Proc Amer Math Soc 90(2):258–262

    Article  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 du cercle a des rotations. Inst Hautes Etudes Sci Publ Math 49:5–233, in French

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  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 spectral analysis of measures. Bull Soc Math France 119:33–90

    Google Scholar 

  105. Hurewicz W (1944) Ergodic theorem without invariant measure. Ann Math 45:192–206

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  108. Ismagilov RS (1987) Application of a group algebra to problems on the tail σ‑algebra of a random walk on a group and to problems on the ergodicity of a skew action. Izv Akad Nauk SSSR Ser Mat 51(4):893–907

    MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  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 and entropy. Ann Probab 11:457–490

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Article  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

    Chapter  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Google Scholar 

  125. Krieger W (1969) On nonsingular transformations of a measure space, vol I, 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 nonsingular transformations of a measure space. Invent Math 15:144–163; Erratum in 26:323–328

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  129. Krieger W (1976) On ergodic flows and isomorphism of factors. Math Ann 223:19–70

    Article  MathSciNet  MATH  Google Scholar 

  130. Kubo I (1969) Quasi-flows. Nagoya Math J 35:1–30

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  135. Mackey GW (1966) Ergodic theory and virtual group. Math Ann 166:187–207

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  139. Méla J-F (1983) Groupes de valeurs propres des systèmes dynamiques et sous‐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 Math 38:179–226

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  145. Mortiss G (2002) Average co‐ordinate entropy. J Aust Math Soc 73:171–186

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  147. Nadkarni MG (1979) On spectra of nonsingular transformations and flows. 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

    Article  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 Sci 13:167–172

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  153. Osikawa M, Hamachi T (1971) On zero type and positive type transformations with 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 invariant measure. Proc Lond Math Soc 3 13:605–612

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  160. Ramsay A (1971) Virtual groups and group actions. Adv Math 6:243–322

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  165. Roy E (2007) Ergodic properties of Poissonian ID processes. Ann Probab 35:551–576

    Article  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 nonsingular transformations. Ergod Theory Dynam Syst 9:759–800

    Article  MathSciNet  MATH  Google Scholar 

  169. Ryzhikov VV (1994) Factorization of an automorphism of a full Boolean algebra into the product of three involutions. Mat Zametki 54(2):79–84,159; in Russian. Translation in: Math Notes 54(1–2):821–824

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  174. Schmidt K (1977) Infinite invariant measures in the circle. Symp Math 21:37–43

    Google Scholar 

  175. Schmidt K (1982) Spectra of ergodic group actions. Isr J Math 41(1–2):151–153

    Article  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  181. Silva CE, Witte D (1992) On quotients of nonsingular actions whose self‐joinings are graphs. Int J Math 5:219–237

    Article  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 to non‐invariant measures. Pac J Math 130:187–193

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  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 of cocycles. Isr J Math 26:84–90

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

Download references

Author information

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

Authors
  1. Alexandre I. Danilenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cesar E. Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag

About this entry

Cite this entry

Danilenko, A.I., Silva, C.E. (2012). Ergodic Theory : Non-singular Transformations . In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_22

Download citation

Publish with us

Policies and ethics

Search

Navigation