Skip to main content
SpringerLink
Log in

Ergodicity and Mixing Properties

  • Reference work entry
Encyclopedia of Complexity and Systems Science

Definition of the Subject

Many physical phenomena in equilibrium can be modeled as measure‐preserving transformations. Ergodic theory is the abstract study of thesetransformations, dealing in particular with their long term average behavior.

One of the basic steps in analyzing a measure‐preserving transformation is to break it down into its simplest possible components. Thesesimplest components are its ergodic components, and on each of these components, the system enjoys the ergodic property: the long-term time average of anymeasurement as the system evolves is equal to the average over the component. Ergodic decomposition gives a precise description of the manner inwhich a system can be split into ergodic components.

A related (stronger) property of a measure‐preserving transformation is mixing. Here one is investigating the correlation betweenthe state of the system at different times. The system is mixing if the states are asymptotically independent: as the times between the...

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

Access this chapter

Log in via an institution

Institutional subscriptions

Abbreviations

Bernoulli shift:

Mathematical abstraction of the scenario in statistics or probability in which one performs repeated independent identical experiments.

Markov chain:

A probability model describing a sequence of observations made at regularly spaced time intervals such that at each time, the probability distribution of the subsequent observation depends only on the current observation and not on prior observations.

Measure-preserving transformation:

A map from a measure space to itself such that for each measurable subset of the space, it has the same measure as its inverse image under the map.

Measure-theoretic entropy:

A non-negative (possibly infinite) real number describing the complexity of a measure‐preserving transformation.

Product transformation:

Given a pair of measure‐preserving transformations: T of X and S of Y, the product transformation is the map of \( { X\times Y } \) given by \( { (T\times S)(x,y)=(T(x),S(y)) } \).

Bibliography

  1. Aaronson J (1997) An Introduction to Infinite Ergodic Theory. AmericanMathematical Society, Providence

    MATH  Google Scholar 

  2. Adams TM, Petersen K (1998) Binomial coefficient multiples of irrationals.Monatsh Math 125:269–278

    MathSciNet  MATH  Google Scholar 

  3. Alpern S (1976) New proofs that weak mixing is generic. Invent Math32:263–278

    MathSciNet  ADS  MATH  Google Scholar 

  4. Avila A, Forni G (2007) Weak-mixing for interval exchange transformations andtranslation flows. Ann Math 165:637–664

    MathSciNet  MATH  Google Scholar 

  5. Bergelson V (1987) Weakly mixing pet. Ergodic Theory Dynam Systems7:337–349

    MathSciNet  MATH  Google Scholar 

  6. Birkhoff GD (1931) Proof of the ergodic theorem. Proc Nat Acad Sci17:656–660

    ADS  Google Scholar 

  7. Birkhoff GD, Smith PA (1924) Structural analysis of surface transformations. J Math 7:345–379

    Google Scholar 

  8. Boltzmann L (1871) Einige allgemeine Sätze über Wärmegleichgewicht. Wiener Berichte 63:679–711

    MATH  Google Scholar 

  9. Boltzmann L (1909) Wissenschaftliche Abhandlungen. Akademie der Wissenschaften, Berlin

    Google Scholar 

  10. Boshernitzan M, Berend D, Kolesnik G (2001) Irrational dilations of Pascal'striangle. Mathematika 48:159–168

    MathSciNet  MATH  Google Scholar 

  11. Bowen R (1975) Equilibrium States and the Ergodic Theory of AnosovDiffeomorphisms. Springer, Berlin

    MATH  Google Scholar 

  12. Bradley RC (2005) Basic properties of strong mixing conditions. A survey andsome open questions. Probab Surv 2:107–144

    MathSciNet  MATH  Google Scholar 

  13. Chacon RV (1969) Weakly mixing transformations which are not strongly mixing.Proc Amer Math Soc 22:559–562

    MathSciNet  MATH  Google Scholar 

  14. Choquet G (1956) Existence des représentations intégrales au moyen des pointsextrémaux dans les cônes convexes. C R Acad Sci Paris 243:699–702

    MathSciNet  MATH  Google Scholar 

  15. Choquet G (1956) Unicité des représentations intégrales au moyen de pointsextrémaux dans les cônes convexes réticulés. C R Acad Sci Paris 243:555–557

    MathSciNet  MATH  Google Scholar 

  16. de la Rue T (2006) 2-fold and 3-fold mixing: why 3-dot-typecounterexamples are impossible in one dimension. Bull Braz Math Soc (NS) 37(4):503–521

    MATH  Google Scholar 

  17. Ehrenfest P, Ehrenfest T (1911) Begriffliche Grundlage der statistischenAuffassung in der Mechanik. No. 4. In: Encyclopädie der mathematischen Wissenschaften. Teubner, Leipzig

    Google Scholar 

  18. Feller W (1950) An Introduction to Probability and its Applications. Wiley,New York

    MATH  Google Scholar 

  19. Friedman NA, Ornstein DS (1970) On isomorphism of weak Bernoullitransformations. Adv Math 5:365–394

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  22. Furstenberg H, Weiss B (1978) The finite multipliers of infinite ergodictransformations. In: The Structure of Attractors in Dynamical Systems. Proc Conf, North Dakota State Univ, Fargo, N.D., 1977. Springer,Berlin

    Google Scholar 

  23. Garsia AM (1965) A simple proof of E. Hopf's maximal ergodic theory. J MathMech 14:381–382

    MathSciNet  MATH  Google Scholar 

  24. Girsanov IV (1958) Spectra of dynamical systems generated by stationaryGaussian processes. Dokl Akad Nauk SSSR 119:851–853

    MathSciNet  MATH  Google Scholar 

  25. Góra P (1994) Properties of invariant measures for piecewise expandingone-dimensional transformations with summable oscillations of derivative. Ergodic Theory Dynam Syst 14:475–492

    Google Scholar 

  26. Halmos PA (1944) In general a measure-preserving transformation is mixing.Ann Math 45:786–792

    MathSciNet  MATH  Google Scholar 

  27. Halmos P (1956) Lectures on Ergodic Theory. Chelsea, NewYork

    MATH  Google Scholar 

  28. Hoffman C, Heicklen D (2002) Rational maps are d‑adic bernoulli. Ann Math 156:103–114

    MathSciNet  MATH  Google Scholar 

  29. Hoffman C, Rudolph DJ (2002) Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann Math 76:79–101

    MathSciNet  Google Scholar 

  30. Host B (1991) Mixing of all orders and pairwise independent joinings ofsystems with singular spectrum. Israel J Math 76:289–298

    MathSciNet  MATH  Google Scholar 

  31. Hu H (2004) Decay of correlations for piecewise smooth maps with indifferentfixed points. Ergodic Theory Dynam Syst 24:495–524

    MATH  Google Scholar 

  32. Kalikow S () Outline of ergodic theory. Notes freely available for download.See http://www.math.uvic.ca/faculty/aquas/kalikow/kalikow.html

  33. Kalikow S (1982) \( { T,T^{-1} } \) transformation is not loosely Bernoulli. Ann Math115:393–409

    MathSciNet  MATH  Google Scholar 

  34. Kalikow S (1984) Twofold mixing implies threefold mixing for rank onetransformations. Ergodic Theory Dynam Syst 2:237–259

    MathSciNet  Google Scholar 

  35. Kamae T (1982) A simple proof of the ergodic theorem using non-standardanalysis. Israel J Math 42:284–290

    MathSciNet  MATH  Google Scholar 

  36. Katok A (1980) Interval exchange transformations and some special flows arenot mixing. Israel J Math 35:301–310

    MathSciNet  MATH  Google Scholar 

  37. Katok A (1980) Smooth non-Bernoulli K-automorphisms. Invent Math61:291–299

    MathSciNet  ADS  MATH  Google Scholar 

  38. Katok A, Hasselblatt B (1995) Introduction to the Modern Theory of DynamicalSystems. Cambridge, Cambridge

    MATH  Google Scholar 

  39. Katznelson Y (1971) Ergodic automorphisms of \( { \mathbb{T}^n } \) are Bernoulli shifts. Israel J Math10:186–195

    MathSciNet  MATH  Google Scholar 

  40. Katznelson Y, Weiss B (1982) A simple proof of some ergodic theorems. IsraelJ Math 42:291–296

    MathSciNet  MATH  Google Scholar 

  41. Keane M, Smorodinsky M (1979) Bernoulli schemes of the same entropy arefinitarily isomorphic. Ann Math 109:397–406

    MathSciNet  MATH  Google Scholar 

  42. Keane MS, Petersen KE (2006) Nearly simultaneous proofs of the ergodic theoremand maximal ergodic theorem. In: Dynamics and Stochastics: Festschrift in Honor of M.S. Keane. Institute of Mathematical Statistics, pp 248–251,Bethesda MD

    Google Scholar 

  43. Kolmogorov AN (1958) New metric invariant of transitive dynamical systems andendomorphisms of Lebesgue spaces. Dokl Russ Acad Sci 119:861–864

    MathSciNet  MATH  Google Scholar 

  44. Koopman BO (1931) Hamiltonian systems and Hilbert space. Proc Nat Acad Sci17:315–218

    ADS  Google Scholar 

  45. Krámli A, Simányi N, Szász D (1991) The K‑property of three billiard balls. Ann Math 133:37–72

    Google Scholar 

  46. Ledrappier F (1978) Un champ Markovien peut être d'entropie nulle etmélangeant. C R Acad Sci Paris, Sér A-B 287:561–563

    MathSciNet  MATH  Google Scholar 

  47. Liverani C (2004) Decay of correlations. Ann Math159:1275–1312

    MathSciNet  MATH  Google Scholar 

  48. Masser DW (2004) Mixing and linear equations over groups in positivecharacteristic. Israel J Math 142:189–204

    MathSciNet  MATH  Google Scholar 

  49. Masur H (1982) Interval exchange transformations and measured foliations. AnnMath 115:169–200

    MathSciNet  MATH  Google Scholar 

  50. McCutcheon R, Quas A (2007) Generalized polynomials and mild mixing systems.Canad J Math (to appear)

    Google Scholar 

  51. Méla X, Petersen K (2005) Dynamical properties of the Pascal adictransformation. Ergodic Theory Dynam Syst 25:227–256

    Google Scholar 

  52. Newton D, Parry W (1966) On a factor automorphism of a normaldynamical system. Ann Math Statist 37:1528–1533

    MathSciNet  MATH  Google Scholar 

  53. Ornstein DS (1970) Bernoulli shifts with the same entropy are isomorphic. AdvMath 4:337–352

    MathSciNet  MATH  Google Scholar 

  54. Ornstein DS (1972) On the root problem in ergodic theory. In: Proceedings ofthe Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), vol II: Probability theory.Univ. California Press, pp 347–356

    Google Scholar 

  55. Ornstein DS (1973) An example of a Kolmogorov automorphism that is not a Bernoulli shift. Adv Math 10:49–62

    MathSciNet  MATH  Google Scholar 

  56. Ornstein DS (1973) A K-automorphism with no square root and Pinsker'sconjecture. Adv Math 10:89–102

    MathSciNet  MATH  Google Scholar 

  57. Ornstein DS (1973) A mixing transformation for which Pinsker's conjecturefails. Adv Math 10:103–123

    MathSciNet  MATH  Google Scholar 

  58. Ornstein DS (1974) Ergodic Theory, Randomness, and Dynamical Systems. YaleUniversity Press, Newhaven

    MATH  Google Scholar 

  59. Ornstein DS, Shields PC (1973) An uncountable family of K‑automorphisms.Adv Math 10:89–102

    MATH  Google Scholar 

  60. Ornstein DS, Weiss B (1973) Geodesic flows are Bernoullian. Israel J Math14:184–198

    MathSciNet  MATH  Google Scholar 

  61. Ornstein DS, Weiss B (1975) Unilateral codings of Bernoulli systems. Israel JMath 21:159–166

    MathSciNet  MATH  Google Scholar 

  62. Oxtoby JC (1952) Ergodic sets. Bull Amer Math Soc58:116–136

    MathSciNet  MATH  Google Scholar 

  63. Parry W (1981) Topics in Ergodic Theory. Cambridge,Cambridge

    MATH  Google Scholar 

  64. Petersen K (1983) Ergodic Theory. Cambridge,Cambridge

    MATH  Google Scholar 

  65. Petersen K, Schmidt K (1997) Symmetric Gibbs measures. Trans Amer Math Soc349:2775–2811

    MathSciNet  MATH  Google Scholar 

  66. Phelps R (1966) Lectures on Choquet's Theorem. Van Nostrand, NewYork

    MATH  Google Scholar 

  67. Pinsker MS (1960) Dynamical systems with completely positive or zero entropy.Soviet Math Dokl 1:937–938

    MathSciNet  MATH  Google Scholar 

  68. Quas A (1996) A C 1 expanding map of the circle which is not weak-mixing. Israel J Math93:359–372

    MathSciNet  MATH  Google Scholar 

  69. Quas A (1996) Non-ergodicity for C 1 expanding maps and g‑measures. Ergodic Theory DynamSystems 16:531–543

    MathSciNet  MATH  Google Scholar 

  70. Rényi A (1957) Representations for real numbers and their ergodic properties.Acta Math Acad Sci Hungar 8:477–493

    Google Scholar 

  71. Riesz F (1938) Some mean ergodic theorems. J Lond Math Soc13:274–278

    MathSciNet  Google Scholar 

  72. Rokhlin VA (1948) A ‘general’ measure-preserving transformation isnot mixing. Dokl Akad Nauk SSSR Ser Mat 60:349–351

    MATH  Google Scholar 

  73. Rokhlin VA (1949) On endomorphisms of compact commutative groups. IzvestiyaAkad Nauk SSSR Ser Mat 13:329–340

    MathSciNet  MATH  Google Scholar 

  74. Rokhlin VA, Sinai Y (1961) Construction and properties of invariant measurablepartitions. Dokl Akad Nauk SSSR 141:1038–1041

    MathSciNet  Google Scholar 

  75. Rudin W (1966) Real and Complex Analysis. McGraw Hill, NewYork

    MATH  Google Scholar 

  76. Rudolph DJ (1990) Fundamentals of Measurable Dynamics. Oxford, OxfordUniversity Press

    MATH  Google Scholar 

  77. Ryzhikov VV (1993) Joinings and multiple mixing of the actions of finite rank.Funct Anal Appl 27:128–140

    MathSciNet  MATH  Google Scholar 

  78. Shields P, Thouvenot J-P (1975) Entropy zero × Bernoulli processes areclosed in the \( { \bar{d} } \)-metric. AnnProbab 3:732–736

    MathSciNet  MATH  Google Scholar 

  79. Simányi N (2003) Proof of the Boltzmann–Sinai ergodic hypothesis fortypical hard disk systems. Invent Math 154:123–178

    Google Scholar 

  80. Simányi N (2004) Proof of the ergodic hypothesis for typical hard ballsystems. Ann Henri Poincaré 5:203–233

    Google Scholar 

  81. Simányi N, Szász D (1999) Hard ball systems are completely hyperbolic. AnnMath 149:35–96

    Google Scholar 

  82. Sinai YG (1959) On the notion of entropy of a dynamical system. Dokl RussAcad Sci 124:768–771

    MathSciNet  MATH  Google Scholar 

  83. Sinai YG (1964) On a weak isomorphism of transformations with invariantmeasure. Mat Sb (NS) 63:23–42

    MathSciNet  Google Scholar 

  84. Sinai YG (1970) Dynamical systems with elastic reflections. Ergodic propertiesof dispersing billiards. Uspehi Mat Nauk 25:141–192

    MathSciNet  MATH  Google Scholar 

  85. Sinai Y (1976) Introduction to Ergodic Theory. Princeton, Princeton,translation of the 1973 Russian original

    MATH  Google Scholar 

  86. Sinai YG, Chernov NI (1987) Ergodic properties of some systems oftwo-dimensional disks and three-dimensional balls. Uspekhi Mat Nauk 42:153–174, 256

    MathSciNet  Google Scholar 

  87. Smorodinsky M (1971) A partition on a Bernoulli shift which is not weaklyBernoulli. Math Systems Th 5:201–203

    MathSciNet  MATH  Google Scholar 

  88. Veech W (1982) Gauss measures for transformations on the space on intervalexchange maps. Ann Math 115:201–242

    MathSciNet  MATH  Google Scholar 

  89. Vershik A (1974) A description of invariant measures for actions of certaininfinite-dimensional groups. Soviet Math Dokl 15:1396–1400

    MATH  Google Scholar 

  90. Vershik A (1981) Uniform algebraic approximation of shift and multiplicationoperators. Soviet Math Dokl 24:97–100

    MATH  Google Scholar 

  91. von Neumann J (1932) Proof of the quasi-ergodic hypothesis. Proc Nat Acad SciUSA 18:70–82

    ADS  Google Scholar 

  92. Walters P (1982) An Introduction to Ergodic Theory. Springer,Berlin

    MATH  Google Scholar 

  93. Williams D (1991) Probability with Martingales. Cambridge,Cambridge

    MATH  Google Scholar 

  94. Young LS (1995) Ergodic theory of differentiable dynamical systems. In: Realand Complex Dynamical Systems (Hillerød, 1993). Kluwer, Dordrecht, pp 293–336

    Google Scholar 

  95. Young LS (1998) Statistical properties of dynamical systems with somehyperbolicity. Ann Math 147:585–650

    MATH  Google Scholar 

  96. Yuzvinskii SA (1967) Metric automorphisms with a simple spectrum. Soviet MathDokl 8:243–245

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, Victoria, Canada

    Anthony Quas

Authors
  1. Anthony Quas
    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

Quas, A. (2009). Ergodicity and Mixing Properties. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_175

Download citation

Publish with us

Policies and ethics

Search

Navigation