Skip to main content
SpringerLink
Log in

Ergodic Theorems

  • Reference work entry
Book cover Encyclopedia of Complexity and Systems Science

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 3,499.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
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

Abbreviations

Dynamical system:

in its broadest sense, any set X, with a map \( { T\colon X \to X } \). Theclassical example is: X is a set whose points are the states of some physical systemand the state x is succeeded by the state Tx after one unit of time.

Iteration:

repeated applications of the map T above to arrive at the state of the systemafter n units of time.

Orbit of x :

the forward images \( { x, Tx, T^{2}X \ldots } \) of \( { x \in X } \) underiteration of T. When T is invertible one may consider the forward, backward ortwo-sided orbit of x.

Automorphism:

a dynamical system \( { T\colon X \to X } \), where X is a measure spaceand T is an invertible map preserving measure.

Ergodic average:

if f is a function on X let \( { A_{n}f(x) =n^{-1}\sum_{i=0}^{n-1} f(T^{i}x) } \); the average of the values of f over thefirst n points in the orbit of x.

Ergodic theorem:

an assertion that ergodic averages converge in some sense.

Mean ergodic theorem:

an assertion that ergodic averages converge with respect to some normon a space of functions.

Pointwise ergodic theorem:

an assertion that ergodic averages \( { A_{n}f(x) } \) converge for some or all\( { x \in X } \), usually for a.e. x.

Stationary process:

a sequence \( { (X_{1}, X_{2}, \ldots) } \) of random variables (real orcomplex‐valued measurable functions) on a probability space whose joint distributions areinvariant under shifting \( { (X_{1}, X_{2}, \ldots) } \) to \( { (X_{2}, X_{3}, \ldots) } \).

Uniform distribution:

a sequence \( { \{x_{n}\} } \) in \( { [0,1] } \) is uniformly distributed if for eachinterval \( { I \subset[0,1] } \), the time it spends in I is asymptotically proportional tothe length of I.

Maximal inequality:

an inequality which allows one to bound the pointwise oscillation ofa sequence of functions. An essential tool for proving pointwise ergodic theorems.

Operator:

any linear operator U on a vector space of functions on X, for exampleone arising from a dynamical system T by setting \( { Uf(x) = f(Tx) } \). More generally anylinear transformation on a real or complex vector space.

Positive contraction:

an operator T on a space of functions endowed with a norm\( { \|\cdot\| } \) such that T maps positive functions to positive functions and \( { \|Tf\| \leq|f| } \).

Bibliography

  1. Akcoglu M, Bellow A, Jones RL, Losert V, Reinhold–Larsson K, Wierdl M (1996) The strongsweeping out property for lacunary sequences, Riemann sums, convolution powers, and relatedmatters. Ergodic Theory Dynam Systems 16(2):207–253

    Google Scholar 

  2. Akcoglu M, Jones RL, Rosenblatt JM (2000) The worst sums in ergodic theory. MichiganMath J 47(2):265–285

    MathSciNet  MATH  Google Scholar 

  3. Akcoglu MA (1975) A pointwise ergodic theorem in \( { L\sb{p} } \)-spaces. Canad J Math27(5):1075–1082

    MathSciNet  MATH  Google Scholar 

  4. Akcoglu MA, Chacon RV (1965) A convexity theorem for positive operators. ZWahrsch Verw Gebiete 3:328–332 (1965)

    MathSciNet  MATH  Google Scholar 

  5. Akcoglu MA, Chacon RV (1970) A local ratio theorem. Canad J Math 22:545–552

    MathSciNet  MATH  Google Scholar 

  6. Akcoglu MA, del Junco A (1975) Convergence of averages of point transformations. ProcAmer Math Soc 49:265–266

    MathSciNet  MATH  Google Scholar 

  7. Akcoglu MA, Kopp PE (1977) Construction of dilations of positive \( { L\sb{p} } \)-contractions.Math Z 155(2):119–127

    MathSciNet  MATH  Google Scholar 

  8. Akcoglu MA, Krengel U (1981) Ergodic theorems for superadditive processes. J Reine AngewMath 323:53–67

    MathSciNet  MATH  Google Scholar 

  9. Akcoglu MA, Sucheston L (1978) A ratio ergodic theorem for superadditive processes. ZWahrsch Verw Gebiete 44(4):269–278

    MathSciNet  MATH  Google Scholar 

  10. Alaoglu L, Birkhoff G (1939) General ergodic theorems. Proc Nat Acad Sci USA25:628–630

    MathSciNet  ADS  Google Scholar 

  11. Assani I (2000) Multiple return times theorems for weakly mixing systems. Ann Inst HPoincaré Probab Statist 36(2):153–165

    MathSciNet  ADS  MATH  Google Scholar 

  12. Assani I (2003) Wiener–Wintner ergodic theorems. World Scientific Publishing Co. Inc.,River Edge, NJ

    Google Scholar 

  13. Assani I, Lesigne E, Rudolph D (1995) Wiener–Wintner return‐times ergodic theorem.Israel J Math 92(1–3):375–395

    MathSciNet  MATH  Google Scholar 

  14. Assani I, Buczolich Z, Mauldin RD (2005) An \( { L\sp 1 } \) counting problem in ergodictheory. J Anal Math 95:221–241

    MathSciNet  MATH  Google Scholar 

  15. Auslander L, Green L, Hahn F (1963) Flows on homogeneous spaces. With the assistance ofMarkus L, Massey W and an appendix by Greenberg L Annals of Mathematics Studies, No 53.Princeton University Press, Princeton, NJ

    Google Scholar 

  16. Banach S (1993) Théorie des opérations linéaires. Éditions Jacques Gabay,Sceaux, reprint of the 1932 original

    Google Scholar 

  17. Bellow A (1983) On “bad universal” sequences in ergodic theory II. In: Belley JM, Dubois J, Morales P (eds) Measure theory and its applications. Lecture Notes in Math, vol 1033. Springer, Berlin, pp 74–78

    Google Scholar 

  18. Bellow A (1999) Transference principles in ergodic theory. In: Christ M, Kenig CE, Sadowsky C (eds) Harmonic analysis and partial differential equations, Chicago Lectures in Math. Univ Chicago Press, Chicago, pp 27–39

    Google Scholar 

  19. Bellow A, Calderón A (1999) A weak-type inequality for convolution products. In: Christ M, Kenig CE, Sadowsky C (eds) Harmonic analysis and partial differential equations, Chicago Lectures in Math. Univ Chicago Press, Chicago, pp 41–48

    Google Scholar 

  20. Bellow A, Jones R (eds) (1991) Almost everywhere convergence, II. Academic Press, Boston

    MATH  Google Scholar 

  21. Bellow A, Losert V (1985) The weighted pointwise ergodic theorem and the individualergodic theorem along subsequences. Trans Amer Math Soc 288(1):307–345

    MathSciNet  MATH  Google Scholar 

  22. Bellow A, Jones R, Rosenblatt J (1989) Almost everywhere convergence of powers. In: Edgar GA, Sucheston L (eds) Almost everywhere convergence. Academic, Boston, pp 99–120

    Google Scholar 

  23. Bellow A, Jones R, Rosenblatt J (1990) Convergence for moving averages. Ergodic TheoryDynam Systems 10(1):43–62

    MathSciNet  MATH  Google Scholar 

  24. Bellow A, Jones RL, Rosenblatt J (1992) Almost everywhere convergence of weightedaverages. Math Ann 293(3):399–426

    MathSciNet  MATH  Google Scholar 

  25. Bellow A, Jones R, Rosenblatt J (1994) Almost everywhere convergence of convolutionpowers. Ergodic Theory Dynam Systems 14(3):415–432

    MathSciNet  MATH  Google Scholar 

  26. Bergelson V (1987) Weakly mixing PET. Ergodic Theory Dynam Systems 7(3):337–349

    MathSciNet  MATH  Google Scholar 

  27. Bergelson V (1996) Ergodic Ramsey theory—an update. In: Pollicott M, Schmidt K (eds) Ergodic theory of \( { Z\sp d } \) actions. London Math Soc Lecture Note Ser, vol 228. Cambridge Univ Press, Cambridge, pp 1–61

    Google Scholar 

  28. Bergelson V (2006) Combinatorial and Diophantine applications of ergodic theory. In: Hasselblatt B, Katok A (eds) Handbook of dynamical systems, vol 1B, Appendix A by Leibman A, Appendix B by Quas A, Wierdl M. Elsevier, Amsterdam, pp 745–869

    Google Scholar 

  29. Bergelson V (2007) Some historical remarks and modern questions around the ergodictheorem. Internat Math Nachrichten 205:1–10

    MATH  Google Scholar 

  30. Bergelson V, Leibman A (2002) A nilpotent Roth theorem. Invent Math 147(2):429–470

    MathSciNet  ADS  MATH  Google Scholar 

  31. Bergelson V, Leibman A (2004) Failure of the Roth theorem for solvable groups ofexponential growth. Ergodic Theory Dynam Systems 24(1):45–53

    MathSciNet  MATH  Google Scholar 

  32. Bergelson V, Leibman A (2007) Distribution of values of bounded generalizedpolynomials. Acta Math 198(2):155–230

    MathSciNet  MATH  Google Scholar 

  33. Berkson E, Bourgain J, Gillespie TA (1991) On the almost everywhere convergence ofergodic averages for power‐bounded operators on \( { L\sp p } \)-subspaces. Integral Equ Operator Theory14(5):678–715

    MathSciNet  MATH  Google Scholar 

  34. Billingsley P (1965) Ergodic theory and information. Wiley, New York

    MATH  Google Scholar 

  35. Birkhoff GD (1931) Proof of the ergodic theorem. Proc Nat Acad Sci USA 17:656–660

    ADS  Google Scholar 

  36. Bishop E (1966) An upcrossing inequality with applications. Michigan Math J 13:1–13

    MathSciNet  MATH  Google Scholar 

  37. Bishop E (1967/1968) A constructive ergodic theorem. J Math Mech 17:631–639

    Google Scholar 

  38. Blum JR, Hanson DL (1960) On the mean ergodic theorem for subsequences. Bull Amer MathSoc 66:308–311

    MathSciNet  MATH  Google Scholar 

  39. Bourgain J (1987) On pointwise ergodic theorems for arithmetic sets. C R Acad Sci ParisSér I Math 305(10):397–402

    MathSciNet  MATH  Google Scholar 

  40. Bourgain J (1988) Almost sure convergence and bounded entropy. Israel J Math63(1):79–97

    MathSciNet  MATH  Google Scholar 

  41. Bourgain J (1988) An approach to pointwise ergodic theorems. In: Lindenstrauss J, Milman VD (eds) Geometric aspects of functional analysis (1986/87). Lecture Notes in Math, vol 1317. Springer, Berlin, pp 204–223

    Google Scholar 

  42. Bourgain J (1988) On the maximal ergodic theorem for certain subsets of the integers.Israel J Math 61(1):39–72

    MathSciNet  MATH  Google Scholar 

  43. Bourgain J (1988) On the pointwise ergodic theorem on \( { L\sp p } \) for arithmetic sets.Israel J Math 61(1):73–84

    MathSciNet  MATH  Google Scholar 

  44. Bourgain J (1988) Temps de retour pour les systèmes dynamiques. C R Acad Sci ParisSér I Math 306(12):483–485

    MathSciNet  MATH  Google Scholar 

  45. Bourgain J (1988) Temps de retour pour les systèmes dynamiques. C R Acad Sci ParisSér I Math 306(12):483–485

    MathSciNet  MATH  Google Scholar 

  46. Bourgain J (1989) Almost sure convergence in ergodic theory. In: Edgar GA, Sucheston L (eds) Almost everywhere convergence. Academic, Boston, pp 145–151

    Google Scholar 

  47. Bourgain J (1989) Pointwise ergodic theorems for arithmetic sets. Inst Hautes ÉtudesSci Publ Math (69):5–45, with an appendix by the author, Furstenberg H, Katznelson Y, Ornstein DS

    Google Scholar 

  48. Bourgain J (1990) Double recurrence and almost sure convergence. J Reine Angew Math404:140–161

    MathSciNet  MATH  Google Scholar 

  49. Breiman L (1957) The individual ergodic theorem of information theory. Ann Math Statist28:809–811

    MathSciNet  MATH  Google Scholar 

  50. Breiman L (1968) Probability. Addison–Wesley, Reading

    MATH  Google Scholar 

  51. Brunel A, Keane M (1969) Ergodic theorems for operator sequences. ZWahrsch Verw Gebiete 12:231–240

    MathSciNet  MATH  Google Scholar 

  52. Burkholder DL (1962) Semi–Gaussian subspaces. Trans Amer Math Soc 104:123–131

    MathSciNet  MATH  Google Scholar 

  53. Calderon AP (1953) A general ergodic theorem. Ann Math (2) 58:182–191

    MathSciNet  MATH  Google Scholar 

  54. Calderón AP (1968) Ergodic theory and translation‐invariant operators. Proc NatlAcad Sci USA 59:349–353

    Google Scholar 

  55. Calderón AP (1968) Ergodic theory and translation‐invariant operators. Proc NatlAcad Sci USA 59:349–353

    Google Scholar 

  56. Calderon AP, Zygmund A (1952) On the existence of certain singular integrals. ActaMath 88:85–139

    MathSciNet  MATH  Google Scholar 

  57. Chacon RV (1962) Identification of the limit of operator averages. J Math Mech11:961–968

    MathSciNet  MATH  Google Scholar 

  58. Chacon RV (1963) Convergence of operator averages. In: Wright FB (ed) Ergodic theory. Academic, New York, pp 89–120

    Google Scholar 

  59. Chacon RV (1963) Linear operators in \( { L\sb{1} } \). In: Wright FB (ed) Ergodic theory. Academic, New York, pp 75–87

    Google Scholar 

  60. Chacon RV (1964) A class of linear transformations. Proc Amer Math Soc 15:560–564

    MathSciNet  MATH  Google Scholar 

  61. Chacon RV, Ornstein DS (1960) A general ergodic theorem. Illinois J Math 4:153–160

    MathSciNet  MATH  Google Scholar 

  62. Conze JP, Lesigne E (1984) Théorèmes ergodiques pour des mesures diagonales. BullSoc Math France 112(2):143–175

    MathSciNet  MATH  Google Scholar 

  63. Conze JP, Lesigne E (1988) Sur un théorème ergodique pour des mesures diagonales. In: Probabilités, Publ Inst Rech Math Rennes, vol 1987. Univ Rennes I, Rennes, pp 1–31

    Google Scholar 

  64. Cotlar M (1955) On ergodic theorems. Math Notae 14:85–119 (1956)

    MathSciNet  Google Scholar 

  65. Cotlar M (1955) A unified theory of Hilbert transforms and ergodic theorems. Rev MatCuyana 1:105–167 (1956)

    MathSciNet  Google Scholar 

  66. Day M (1942) Ergodic theorems for Abelian semigroups. Trans Amer Math Soc 51:399–412

    MathSciNet  Google Scholar 

  67. Demeter C, Lacey M, Tao T, Thiele C (2008) Breaking the duality in the return times theorem. Duke Math J 143(2):281–355

    MathSciNet  MATH  Google Scholar 

  68. Derrien JM, Lesigne E (1996) Un théorème ergodique polynomial ponctuel pour lesendomorphismes exacts et les K‑systèmes. Ann Inst H Poincaré Probab Statist 32(6):765–778

    MathSciNet  MATH  Google Scholar 

  69. Derriennic Y (2006) Some aspects of recent works on limit theorems in ergodic theorywith special emphasis on the “central limit theorem”. Discrete Contin Dyn Syst 15(1):143–158

    MathSciNet  MATH  Google Scholar 

  70. Derriennic Y (2006) Some aspects of recent works on limit theorems in ergodic theorywith special emphasis on the “central limit theorem”. Discrete Contin Dyn Syst 15(1):143–158

    MathSciNet  MATH  Google Scholar 

  71. Doob JL (1938) Stochastic processes with an integral‐valued parameter. Trans Amer MathSoc 44(1):87–150

    MathSciNet  Google Scholar 

  72. Dunford N (1951) An individual ergodic theorem for non‐commutative transformations.Acta Sci Math Szeged 14:1–4

    MathSciNet  MATH  Google Scholar 

  73. Dunford N, Schwartz J (1955) Convergence almost everywhere of operator averages. ProcNatl Acad Sci USA 41:229–231

    MathSciNet  ADS  MATH  Google Scholar 

  74. Dunford N, Schwartz JT (1988) Linear operators, Part I. Wiley Classics Library, Wiley,New York. General theory, with the assistance of Bade WG, Bartle RG, Reprint of the 1958 original,A Wiley–Interscience Publication

    Google Scholar 

  75. Durand S, Schneider D (2003) Random ergodic theorems and regularizing random weights.Ergodic Theory Dynam Systems 23(4):1059–1092

    MathSciNet  MATH  Google Scholar 

  76. Eberlein WF (1949) Abstract ergodic theorems and weak almost periodic functions. TransAmer Math Soc 67:217–240

    MathSciNet  MATH  Google Scholar 

  77. Edgar G, Sucheston L (eds) (1989) Almost everywhere convergence. Academic Press,Boston

    MATH  Google Scholar 

  78. Emerson WR (1974) The pointwise ergodic theorem for amenable groups. Amer J Math96:472–487

    MathSciNet  MATH  Google Scholar 

  79. Feldman J (2007) A ratio ergodic theorem for commuting, conservative, invertibletransformations with quasi‐invariant measure summed over symmetric hypercubes. Ergodic TheoryDynam Systems 27(4):1135–1142

    MathSciNet  MATH  Google Scholar 

  80. Foguel SR (1969) The ergodic theory of Markov processes. In: Van NostrandMathematical Studies, No 21. Van Nostrand Reinhold, New York

    Google Scholar 

  81. Frantzikinakis N, Kra B (2005) Convergence of multiple ergodic averages for somecommuting transformations. Ergodic Theory Dynam Systems 25(3):799–809

    MathSciNet  MATH  Google Scholar 

  82. Frantzikinakis N, Kra B (2005) Polynomial averages converge to the product ofintegrals. Israel J Math 148:267–276

    MathSciNet  MATH  Google Scholar 

  83. Frantzikinakis N, Kra B (2006) Ergodic averages for independent polynomials andapplications. J London Math Soc (2) 74(1):131–142

    MathSciNet  Google Scholar 

  84. Furstenberg H (1967) Disjointness in ergodic theory, minimal sets, and a problem inDiophantine approximation. Math Systems Theory 1:1–49

    MathSciNet  MATH  Google Scholar 

  85. Furstenberg H (1977) Ergodic behavior of diagonal measures and a theorem of Szemerédion arithmetic progressions. J Analyse Math 31:204–256

    MathSciNet  MATH  Google Scholar 

  86. Furstenberg H (1981) Recurrence in ergodic theory and combinatorial number theory.Princeton University Press, Princeton, NJ, m B Porter Lectures

    MATH  Google Scholar 

  87. Furstenberg H, Kesten H (1960) Products of random matrices. Ann Math Statist31:457–469

    MathSciNet  MATH  Google Scholar 

  88. Furstenberg H, Weiss B (1996) A mean ergodic theorem for \( (1/N)\sum\sp N\sb {n=1}f(T\sp nx)g(T\sp {n\sp 2}x) \). In: Bergelson V, March P, Rosenblatt J (eds) Convergence in ergodic theory and probability. Ohio State Univ Math Res Inst Publ, vol 5. de Gruyter, Berlin, pp 193–227

    Google Scholar 

  89. Garsia AM (1970) Topics in almost everywhere convergence. In: Lectures in Advanced Mathematics, vol 4. Markham, Chicago

    Google Scholar 

  90. Glasner E (2003) Ergodic theory via joinings, Mathematical Surveys and Monographs, vol 101. American Mathematical Society, Providence

    Google Scholar 

  91. Gowers WT (2001) A new proof of Szemerédi's theorem. Geom Funct Anal 11(3):465–588

    MathSciNet  MATH  Google Scholar 

  92. Green B, Tao T (2004) The primes contain arbitrarily large arithmetic progressions.http://arxivorg/abs/mathNT/0404188

  93. Greenleaf FP (1973) Ergodic theorems and the construction of summing sequences inamenable locally compact groups. Comm Pure Appl Math 26:29–46

    MathSciNet  MATH  Google Scholar 

  94. Greenleaf FP, Emerson WR (1974) Group structure and the pointwise ergodic theorem forconnected amenable groups. Adv Math 14:153–172

    MathSciNet  MATH  Google Scholar 

  95. Guivarc'h Y (1969) Généralisation d'un théorème de von Neumann. C R Acad SciParis Sér A-B 268:A1020–A1023

    MathSciNet  Google Scholar 

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

    MathSciNet  ADS  MATH  Google Scholar 

  97. Halmos PR (1949) A nonhomogeneous ergodic theorem. Trans Amer Math Soc 66:284–288

    MathSciNet  MATH  Google Scholar 

  98. Halmos PR (1960) Lectures on ergodic theory. Chelsea, New York

    MATH  Google Scholar 

  99. Hammersley JM, Welsh DJA (1965)First‐passage percolation, subadditive processes, stochasticnetworks, and generalized renewal theory. In: Proc Internat ResSemin. Statist Lab, University of California, Berkeley. Springer, NewYork, pp 61–110

    Google Scholar 

  100. Hopf E (1937) Ergodentheorie. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 5. Springer, Berlin

    Google Scholar 

  101. Hopf E (1954) The general temporally discrete Markoff process. J Rational Mech Anal3:13–45

    MathSciNet  MATH  Google Scholar 

  102. Host B, Kra B (2001) Convergence of Conze–Lesigne averages. Ergodic Theory DynamSystems 21(2):493–509

    MathSciNet  MATH  Google Scholar 

  103. Host B, Kra B (2005) Convergence of polynomial ergodic averages. Israel J Math149:1–19

    MathSciNet  MATH  Google Scholar 

  104. Host B, Kra B (2005) Nonconventional ergodic averages and nilmanifolds. Ann Math(2) 161(1):397–488

    MathSciNet  Google Scholar 

  105. Hurewicz W (1944) Ergodic theorem without invariant measure. Ann Math (2) 45:192–206

    MathSciNet  MATH  Google Scholar 

  106. Tulcea AI (1964) Ergodic preperties of positive isometries. Bull AMS 70:366–371

    MATH  Google Scholar 

  107. Ivanov VV (1996) Geometric properties of monotone functions and the probabilities ofrandom oscillations. Sibirsk Mat Zh 37(1):117–15

    MathSciNet  Google Scholar 

  108. Ivanov VV (1996) Oscillations of averages in the ergodic theorem. Dokl Akad Nauk347(6):736–738

    MathSciNet  Google Scholar 

  109. Jewett RI (1969/1970) The prevalence of uniquely ergodic systems. J Math Mech19:717–729

    Google Scholar 

  110. Jones R, Rosenblatt J, Tempelman A (1994) Ergodic theorems for convolutions of a measure on a group. Illinois J Math 38(4):521–553

    MathSciNet  MATH  Google Scholar 

  111. Jones RL (1987) Necessary and sufficient conditions for a maximal ergodic theoremalong subsequences. Ergodic Theory Dynam Systems 7(2):203–210

    MathSciNet  MATH  Google Scholar 

  112. Jones RL (1993) Ergodic averages on spheres. J Anal Math 61:29–45

    MathSciNet  MATH  Google Scholar 

  113. Jones RL, Wierdl M (1994) Convergence and divergence of ergodic averages. ErgodicTheory Dynam Systems 14(3):515–535

    MathSciNet  MATH  Google Scholar 

  114. Jones RL, Olsen J, Wierdl M (1992) Subsequence ergodic theorems for \( { L\sp p } \)contractions. Trans Amer Math Soc 331(2):837–850

    MathSciNet  MATH  Google Scholar 

  115. Jones RL, Kaufman R, Rosenblatt JM, Wierdl M (1998) Oscillation in ergodic theory.Ergodic Theory Dynam Systems 18(4):889–935

    MathSciNet  MATH  Google Scholar 

  116. Jones RL, Lacey M, Wierdl M (1999) Integer sequences with big gaps and the pointwiseergodic theorem. Ergodic Theory Dynam Systems 19(5):1295–1308

    MathSciNet  MATH  Google Scholar 

  117. Jones RL, Rosenblatt JM, Wierdl M (2001) Oscillation inequalities for rectangles. ProcAmer Math Soc 129(5):1349–1358 (electronic)

    MathSciNet  MATH  Google Scholar 

  118. JonesRL, Rosenblatt JM, Wierdl M (2003) Oscillation in ergodic theory: higherdimensional results. Israel J Math 135:1–27

    Google Scholar 

  119. del Junco A, Rosenblatt J (1979) Counterexamples in ergodic theory and number theory.Math Ann 245(3):185–197

    MathSciNet  MATH  Google Scholar 

  120. Kac M (1947) On the notion of recurrence in discrete stochastic processes. Bull AmerMath Soc 53:1002–1010

    MathSciNet  MATH  Google Scholar 

  121. Kachurovskii AG (1996) Rates of convergence in ergodic theorems. Uspekhi Mat Nauk51(4(310)):73–124

    MathSciNet  Google Scholar 

  122. Kachurovskii AG (1996) Spectral measures and convergence rates in the ergodictheorem. Dokl Akad Nauk 347(5):593–596

    MathSciNet  Google Scholar 

  123. Kakutani S (1940) Ergodic theorems and the Markoff process with a stable distribution.Proc Imp Acad Tokyo 16:49–54

    MathSciNet  Google Scholar 

  124. Kalikow S, Weiss B (1999) Fluctuations of ergodic averages. In: Proceedings of the Conference on Probability, Ergodic Theory, and Analysis, vol 43. pp 480–488

    Google Scholar 

  125. Kamae T (1982) A simple proof of the ergodic theorem using nonstandard analysis.Israel J Math 42(4):284–290

    MathSciNet  MATH  Google Scholar 

  126. Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems,Encyclopedia of Mathematics and its Applications, vol 54. Cambridge University Press, Cambridge,with a supplementary chapter by Katok A and Mendoza L

    Google Scholar 

  127. Katznelson Y, Weiss B (1982) A simple proof of some ergodic theorems. Israel J Math42(4):291–296

    MathSciNet  MATH  Google Scholar 

  128. Kieffer JC (1975) A generalized Shannon–McMillan theorem for the action of an amenablegroup on a probability space. Ann Probability 3(6):1031–1037

    MathSciNet  Google Scholar 

  129. Kingman JFC (1968) The ergodic theory of subadditive stochastic processes. J RoyStatist Soc Ser B 30:499–510

    MathSciNet  MATH  Google Scholar 

  130. Kingman JFC (1976) Subadditive processes. In: École d'Été de Probabilités deSaint–Flour, V–1975. Lecture Notes in Math, vol 539. Springer, Berlin, pp 167–223

    Google Scholar 

  131. Koopman B (1931) Hamiltonian systems and transformations in hilbert spaces. Proc NatlAcad Sci USA 17:315–318

    ADS  Google Scholar 

  132. Krantz SG, Parks HR (1999) The geometry of domains in space. Birkhäuser AdvancedTexts: Basler Lehrbücher. Birkhäuser Boston Inc, Boston, MA

    Google Scholar 

  133. Krengel U (1971) On the individual ergodic theorem for subsequences. Ann Math Statist42:1091–1095

    MathSciNet  MATH  Google Scholar 

  134. Krengel U (1978/79) On the speed of convergence in the ergodic theorem. Monatsh Math86(1):3–6

    Google Scholar 

  135. Krengel U (1985) Ergodic theorems. In: de Gruyter studies in mathematics, vol 6. de Gruyter, Berlin, with a supplement by Antoine Brunel

    Google Scholar 

  136. Krengel U, Lin M, Wittmann R (1990) A limit theorem for order preserving nonexpansiveoperators in \( { L\sb 1 } \). Israel J Math 71(2):181–191

    MathSciNet  MATH  Google Scholar 

  137. Krieger W (1972) On unique ergodicity. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol II. Probability Theory, University of California Press, Berkeley, pp 327–346

    Google Scholar 

  138. Kryloff N, Bogoliouboff N (1937) La théorie générale de la mesure dans sonapplication à l'étude des systèmes dynamiques de la mécanique non linéaire. Ann Math(2) 38(1):65–113

    MathSciNet  Google Scholar 

  139. Kuipers L, Niederreiter H (1974) Uniform distribution of sequences.Wiley, New York, Pure and Applied Mathematics

    MATH  Google Scholar 

  140. Lamperti J (1958) On the isometries of certain function‐spaces. Pacific J Math8:459–466

    MathSciNet  MATH  Google Scholar 

  141. Leibman A (2005) Convergence of multiple ergodic averages along polynomials of severalvariables. Israel J Math 146:303–315

    MathSciNet  MATH  Google Scholar 

  142. Leibman A (2005) Pointwise convergence of ergodic averages for polynomial actions of\( { \mathbb{Z}\sp d } \) by translations on a nilmanifold. Ergodic Theory Dynam Systems 25(1):215–225

    MathSciNet  MATH  Google Scholar 

  143. Lemańczyk M, Lesigne E, Parreau F, Volný D, Wierdl M

    Google Scholar 

  144. Lesigne E (1989) Théorèmes ergodiques pour une translation sur un nilvariété.Ergodic Theory Dynam Systems 9(1):115–126

    MathSciNet  MATH  Google Scholar 

  145. Lindenstrauss E (1999) Pointwise theorems for amenable groups. Electron Res AnnouncAmer Math Soc 5:82–90

    MathSciNet  MATH  Google Scholar 

  146. Loomis LH (1946) A note on the Hilbert transform. Bull Amer Math Soc 52:1082–1086

    MathSciNet  MATH  Google Scholar 

  147. Lorch ER (1939) Means of iterated transformations in reflexive vector spaces. BullAmer Math Soc 45:945–947

    MathSciNet  Google Scholar 

  148. Mauldin D, Buczolich Z (2005) Concepts behind divergent ergodic averages along the squares. In: Assani I (ed) Ergodic theory and related fields. Contemp Math, vol 430. Amer Math Soc, Providence, pp 41–56

    Google Scholar 

  149. McMillan B (1953) The basic theorems of information theory. Ann Math Statistics24:196–219

    MathSciNet  MATH  Google Scholar 

  150. Merlevède F, Peligrad M, Utev S (2006) Recent advances in invariance principlesfor stationary sequences. Probab Surv 3:1–36

    Google Scholar 

  151. von Neumann J (1932) Proof of the quasi‐ergodic hypothesis. Proc Natl Acad Sci USA18:70–82

    ADS  Google Scholar 

  152. Neveu J (1961) Sur le théorème ergodique ponctuel. C R Acad Sci Paris252:1554–1556

    MathSciNet  MATH  Google Scholar 

  153. Neveu J (1965) Mathematical foundations of the calculus of probability. Translated byAmiel Feinstein, Holden‐Day, San Francisco

    MATH  Google Scholar 

  154. Nevo A (1994) Harmonic analysis and pointwise ergodic theorems for noncommutingtransformations. J Amer Math Soc 7(4):875–902

    MathSciNet  MATH  Google Scholar 

  155. Nevo A (2006) Pointwise ergodic theorems for actions of groups. In: Hasselblatt B, Katok A (eds) Handbook of dynamical systems vol 1B. Elsevier, Amsterdam, pp 871–982

    Google Scholar 

  156. Nevo A, Stein EM (1994) A generalization of Birkhoff's pointwise ergodic theorem. ActaMath 173(1):135–154

    MathSciNet  MATH  Google Scholar 

  157. Nguyen XX (1979) Ergodic theorems for subadditive spatial processes. Z Wahrsch VerwGebiete 48(2):159–176

    MathSciNet  MATH  Google Scholar 

  158. Orey S (1971) Lecture notes on limit theorems for Markov chain transitionprobabilities. In: Van Nostrand Reinhold Mathematical Studies, no 34. Van Nostrand Reinhold Co,London

    Google Scholar 

  159. Ornstein D (1970) Bernoulli shifts with the same entropy are isomorphic. Adv Math 4:337–352 (1970)

    MathSciNet  MATH  Google Scholar 

  160. Ornstein D (1971) A remark on the Birkhoff ergodic theorem. Illinois J Math 15:77–79

    MathSciNet  MATH  Google Scholar 

  161. Ornstein D, Weiss B (1983) The Shannon–McMillan–Breiman theorem for a class ofamenable groups. Israel J Math 44(1):53–60

    MathSciNet  MATH  Google Scholar 

  162. Ornstein D, Weiss B (1992) Subsequence ergodic theorems for amenable groups. Israel JMath 79(1):113–127

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  164. Oseledec VI (1968) A multiplicative ergodic theorem. Characteristic Ljapunov,exponents of dynamical systems. Trudy Moskov Mat Obšč 19:179–210

    MathSciNet  Google Scholar 

  165. Oxtoby JC (1952) Ergodic sets. Bull Amer Math Soc 58:116–136

    MathSciNet  MATH  Google Scholar 

  166. Parry W (1969) Ergodic properties of affine transformations and flows on nilmanifolds.Amer J Math 91:757–771

    MathSciNet  MATH  Google Scholar 

  167. Paterson A (1988) Amenability, Mathematical Surveys and Monographs, vol 29. AmericanMathematical Society, Providence, RI

    Google Scholar 

  168. Peck JEL (1951) An ergodic theorem for a noncommutative semigroup of linear operators.Proc Amer Math Soc 2:414–421

    MathSciNet  MATH  Google Scholar 

  169. Pesin JB (1977) Characteristic Ljapunov exponents, and smooth ergodic theory. UspehiMat Nauk 32(4(196)):55–112,287

    MathSciNet  Google Scholar 

  170. Petersen K (1983) Another proof of the existence of the ergodic Hilbert transform.Proc Amer Math Soc 88(1):39–43

    MathSciNet  MATH  Google Scholar 

  171. Petersen K (1989) Ergodic theory, Cambridge Studies in Advanced Mathematics, vol 2.Cambridge University Press, Cambridge

    Google Scholar 

  172. Pitt HR (1942) Some generalizations of the ergodic theorem. Proc Cambridge Philos Soc38:325–343

    MathSciNet  ADS  MATH  Google Scholar 

  173. Poincaré H (1987) Les méthodes nouvelles de la mécanique céleste. Tome I,II, III. Les Grands Classiques Gauthier–Villars. Librairie Scientifique et Technique AlbertBlanchard, Paris

    Google Scholar 

  174. Raghunathan MS (1979) A proof of Oseledec's multiplicative ergodic theorem. Israel JMath 32(4):356–362

    MathSciNet  MATH  Google Scholar 

  175. Renaud PF (1971) General ergodic theorems for locally compact groups. Amer J Math93:52–64

    MathSciNet  MATH  Google Scholar 

  176. Rosenblatt JM, Wierdl M (1995)Pointwise ergodic theorems via harmonic analysis. In: Peterson KE,Salama IA (eds) Ergodic theory and its connections with harmonicanalysis, London Math Soc Lecture Note Ser, vol 205. CambridgeUniversity Press, Cambridge, pp 3–151

    Google Scholar 

  177. Rudolph DJ (1990) Fundamentals of measurable dynamics. In: Fundamentals of measurable dynamics: Ergodic theory on Lebesgue spaces. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York

    Google Scholar 

  178. Rudolph DJ (1994) A joinings proof of Bourgain's return time theorem. Ergodic TheoryDynam Systems 14(1):197–203

    MathSciNet  MATH  Google Scholar 

  179. Rudolph DJ (1998) Fully generic sequences and a multiple‐term return‐times theorem.Invent Math 131(1):199–228

    MathSciNet  ADS  MATH  Google Scholar 

  180. Ruelle D (1982) Characteristic exponents and invariant manifolds in Hilbert space. Ann Math (2) 115(2):243–290

    MathSciNet  Google Scholar 

  181. Ryll–Nardzewski C (1951) On the ergodic theorems, II. Ergodic theory of continuedfractions. Studia Math 12:74–79

    Google Scholar 

  182. Ryzhikov V (1994) Joinings, intertwining operators, factors and mixing properties ofdynamical systems. Russian Acad Sci Izv Math 42:91–114

    MathSciNet  Google Scholar 

  183. Shah NA (1998) Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements. In: Dani SG (ed) Lie groups and ergodic theory. Tata Inst Fund Res Stud Math, vol 14. Tata Inst Fund Res, Bombay, pp 229–271

    Google Scholar 

  184. Shalom Y (1998) Random ergodic theorems, invariant means and unitary representation. In: Dani SG (ed) Lie groups and ergodic theory. Tata Inst Fund Res Stud Math, vol 14. Tata Inst Fund Res, Bombay, pp 273–314

    Google Scholar 

  185. Shannon CE (1948) A mathematical theory of communication. Bell System Tech J27:379–423, 623–656

    MathSciNet  Google Scholar 

  186. Shields PC (1987) The ergodic and entropy theorems revisited. IEEE Trans Inform Theory33(2):263–266

    MathSciNet  MATH  Google Scholar 

  187. Shulman A (1988) Maximal ergodic theorems on groups. Dep Lit NIINTI No. 2184

    Google Scholar 

  188. Sine R (1970) A mean ergodic theorem. Proc Amer Math Soc 24:438–439

    MathSciNet  MATH  Google Scholar 

  189. Smythe RT (1976) Multiparameter subadditive processes. Ann Probability 4(5):772–782

    MathSciNet  MATH  Google Scholar 

  190. Szemerédi E (1975) On sets of integers containing no k elements in arithmeticprogression. Acta Arith 27:199–245

    Google Scholar 

  191. Tao T (2005) The gaussian primes contain arbitrarily shaped constellations.http://arxivorg/abs/math/0501314

  192. Tao T (2008) Norm convergence of multiple ergodic averages for commuting transformations. Ergod Theory Dyn Syst 28(2):657–688

    ADS  MATH  Google Scholar 

  193. Tao T, Ziegler T (2006) The primes contain arbitrarily long polynomial progressions.http://frontmathucdavisedu/06105050

  194. Tempelman A (1992) Ergodic theorems for group actions, Mathematics and itsApplications, vol 78. Kluwer, Dordrecht, informational andthermodynamical aspects, Translated and revised from the 1986 Russian original

    Google Scholar 

  195. Tempel'man AA (1967) Ergodic theorems for general dynamical systems. Dokl Akad NaukSSSR 176:790–793

    MathSciNet  Google Scholar 

  196. Tempel'man AA (1972) Ergodic theorems for general dynamical systems. Trudy Moskov MatObšč 26:95–132

    MathSciNet  MATH  Google Scholar 

  197. Tempel'man AA (1972) A generalization of a certain ergodic theorem of Hopf. TeorVerojatnost i Primenen 17:380–383

    MathSciNet  Google Scholar 

  198. Thouvenot JP (1995) Some properties and applications of joinings in ergodic theory. In: Peterson KE, Salama IA (eds) Ergodic theory and its connections with harmonic analysis, London Math Soc Lecture Note Ser, vol 205. Cambridge University Press, Cambridge, pp 207–235

    Google Scholar 

  199. Walters P (1993) A dynamical proof of the multiplicative ergodic theorem. Trans AmerMath Soc 335(1):245–257

    MathSciNet  MATH  Google Scholar 

  200. Weber M (1998) Entropie métrique et convergence presque partout, Travaux en Cours,vol 58. Hermann, Paris

    MATH  Google Scholar 

  201. Weiss B (2003) Actions of amenable groups. In: Bezuglyi S, Kolyada S (eds) Topics in dynamics and ergodic theory, London Math Soc Lecture Note Ser, vol 310. Cambridge University Press, Cambridge, pp 226–262

    Google Scholar 

  202. Weyl H (1916) Über die Gleichverteilung von Zahlen mod Eins. Math Ann 77(3):313–352

    MathSciNet  MATH  Google Scholar 

  203. Wiener N (1939) The ergodic theorem. Duke Math J 5(1):1–18

    MathSciNet  Google Scholar 

  204. Wiener N, Wintner A (1941) Harmonic analysis and ergodic theory. Amer J Math63:415–426

    MathSciNet  Google Scholar 

  205. Wierdl M (1988) Pointwise ergodic theorem along the prime numbers. Israel J Math64(3):315–336 (1989)

    MathSciNet  Google Scholar 

  206. Wittmann R (1995) Almost everywhere convergence of ergodic averages of nonlinearoperators. J Funct Anal 127(2):326–362

    MathSciNet  MATH  Google Scholar 

  207. Yosida K (1940) An abstract treatment of the individual ergodic theorem. Proc Imp AcadTokyo 16:280–284

    MathSciNet  Google Scholar 

  208. Yosida K (1940) Ergodic theorems of Birkhoff–Khintchine's type. Jap J Math 17:31–36

    MathSciNet  Google Scholar 

  209. Yosida K, Kakutani S (1939) Birkhoff's ergodic theorem and the maximal ergodictheorem. Proc Imp Acad, Tokyo 15:165–168

    MathSciNet  MATH  Google Scholar 

  210. Ziegler T (2005) A non‐conventional ergodic theorem for a nilsystem. Ergodic TheoryDynam Systems 25(4):1357–1370

    MathSciNet  MATH  Google Scholar 

  211. Ziegler T (2007) Universal characteristic factors and Furstenberg averages. J AmerMath Soc 20(1):53–97 (electronic)

    MathSciNet  MATH  Google Scholar 

  212. Zimmer RJ (1976) Ergodic actions with generalized discrete spectrum. Illinois J Math20(4):555–588

    MathSciNet  MATH  Google Scholar 

  213. Zimmer RJ (1976) Extensions of ergodic group actions. Illinois J Math 20(3):373–409

    MathSciNet  MATH  Google Scholar 

  214. Zund JD (2002) George David Birkhoff and John von Neumann: a question of priority andthe ergodic theorems, 1931–1932. Historia Math 29(2):138–156

    MathSciNet  MATH  Google Scholar 

  215. Zygmund A (1951) An individual ergodic theorem for non‐commutative transformations.Acta Sci Math Szeged 14:103–110

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, Canada

    Andrés del Junco

Authors
  1. Andrés del Junco
    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

© 2009 Springer-Verlag

About this entry

Cite this entry

Junco, A. (2009). Ergodic Theorems. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_176

Download citation

Publish with us

Policies and ethics

Search

Navigation