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Measure Preserving Systems

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Measure-preserving systems model processes in equilibrium by transformations on probability spaces or, more generally, measure spaces. They are thebasic objects of study in ergodic theory, a central part of dynamical systems theory. These systems arise from science and technology as well as frommathematics itself, so applications are found in a wide range of areas, such as statistical physics, information theory, celestial mechanics, numbertheory, population dynamics, economics, and biology.

Introduction: The Dynamical Viewpoint

Sometimes introducing a dynamical viewpoint into an apparently static situation can help to make progress on apparently difficult problems. Forexample, equations can be solved and functions optimized by reformulating a given situation as a fixed point problem, which is then addressed byiterating an appropriate mapping. Besides practical applications, this strategy also appears in theoretical settings, for example modern proofs of...

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Abbreviations

Dynamical system:

A set acted upon by an algebraic object. Elements of the set represent all possible states or configurations, and the action represents all possible changes.

Ergodic:

A measure-preserving system is ergodic if it is essentially indecomposable, in the sense that given any invariant measurable set, either the set or its complement has measure 0.

Lebesgue space:

A measure space that is isomorphic with the usual Lebesgue measure space of a subinterval of the set of real numbers, possibly together with countably or finitely many point masses.

Measure:

An assignment of sizes to sets. A measure that takes values only between 0 and 1 assigns probabilities to events.

Stochastic process:

A family of random variables (measurable functions). Such an object represents a family of measurements whose outcomes may be subject to chance.

Subshift, shift space:

A closed shift-invariant subset of the set of infinite sequences with entries from a finite alphabet.

Bibliography

Primary Literature

  1. Adler RL (1973) F‑expansions revisited. Lecture Notes in Math, vol 318. Springer, Berlin

    Google Scholar 

  2. Arnold LK (1968) On σ-finite invariant measures. Z Wahrscheinlichkeitstheorie Verw Geb 9:85–97

    MATH  Google Scholar 

  3. Billingsley P (1978) Ergodic theory and information. Robert E Krieger Publishing Co, Huntington NY, pp xiii,194; Reprint of the 1965 original

    Google Scholar 

  4. Billingsley P (1995) Probability and measure, 3rd edn. Wiley, New York, pp xiv,593

    MATH  Google Scholar 

  5. Brunel A (1966) Sur les mesures invariantes. Z Wahrscheinlichkeitstheorie Verw Geb 5:300–303

    MathSciNet  MATH  Google Scholar 

  6. Calderón AP (1955) Sur les mesures invariantes. C R Acad Sci Paris 240:1960–1962

    Google Scholar 

  7. Carathéodory C (1939) Die Homomorphieen von Somen und die Multiplikation von Inhaltsfunktionen. Annali della R Scuola Normale Superiore di Pisa 8(2):105–130

    Google Scholar 

  8. Carathéodory C (1968) Vorlesungen über Reelle Funktionen, 3rd edn. Chelsea Publishing Co, New York, pp x,718

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  10. Conway JB (1990) A Course in Functional Analysis, vol 96, 2nd edn. Springer, New York, pp xvi,399

    MATH  Google Scholar 

  11. Dowker YN (1955) On measurable transformations in finite measure spaces. Ann Math 62(2):504–516

    MathSciNet  MATH  Google Scholar 

  12. Dowker YN (1956) Sur les applications mesurables. C R Acad Sci Paris 242:329–331

    MathSciNet  MATH  Google Scholar 

  13. Frechet M (1924) Des familles et fonctions additives d’ensembles abstraits. Fund Math 5:206–251

    Google Scholar 

  14. Friedman NA (1970) Introduction to Ergodic Theory. Van Nostrand Reinhold Co., New York, pp v,143

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Gowers WT (2001) Erratum: A new proof of Szemerédi’s theorem. Geom Funct Anal 11(4):869

    MathSciNet  Google Scholar 

  18. Green B, Tao T (2007) The primes contain arbitrarily long arithmetic progressions. arXiv:math.NT/0404188

    Google Scholar 

  19. Hahn H (1933) Über die Multiplikation total-additiver Mengenfunktionen. Annali Scuola Norm Sup Pisa 2:429–452

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  21. Halmos PR (1947) Invariant measures. Ann Math 48(2):735–754

    MathSciNet  MATH  Google Scholar 

  22. Katok A, Hasselblatt B (1995) Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, pp xviii,802

    MATH  Google Scholar 

  23. Hopf E (1932) Theory of measure and invariant integrals. Trans Amer Math Soc 34:373–393

    MathSciNet  Google Scholar 

  24. Hopf E (1937) Ergodentheorie. Ergebnisse der Mathematik und ihrer Grenzgebiete, 1st edn. Springer, Berlin, pp iv,83

    Google Scholar 

  25. Khinchin AI (1949) Mathematical Foundations of Statistical Mechanics. Dover Publications Inc, New York, pp viii,179

    MATH  Google Scholar 

  26. Kolmogorov AN (1956) Foundations of the Theory of Probability. Chelsea Publishing Co, New York, pp viii,84

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  28. Petersen K (1989) Ergodic Theory. Cambridge Studies in Advanced Mathematics, vol 2. Cambridge University Press, Cambridge, pp xii,329

    Google Scholar 

  29. Poincaré H (1987) Les Méthodes Nouvelles de la Mécanique Céleste. Tomes I, II,III. Les Grands Classiques Gauthier-Villars. Librairie Scientifique et Technique Albert Blanchard, Paris

    Google Scholar 

  30. Radjavi H, Rosenthal P (1973) Invariant subspaces, 2nd edn. Springer, Mineola, pp xii,248

    Google Scholar 

  31. Rohlin VA (1952) On the fundamental ideas of measure theory. Amer Math Soc Transl 1952:55

    MathSciNet  Google Scholar 

  32. Rohlin VA (1960) New progress in the theory of transformations with invariant measure. Russ Math Surv 15:1–22

    MathSciNet  ADS  Google Scholar 

  33. Royden HL (1988) Real Analysis, 3rd edn. Macmillan Publishing Company, New York, pp xx,444

    MATH  Google Scholar 

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

    Google Scholar 

  35. Tao T (2006) Szemerédi’s regularity lemma revisited. Contrib Discret Math 1:8–28

    MATH  Google Scholar 

  36. Tao T (2006) Arithmetic progressions and the primes. Collect Math Extra:37–88

    Google Scholar 

  37. von Neumann J (1932) Einige Sätze über messbare Abbildungen. Ann Math 33:574–586

    Google Scholar 

  38. Wright FB (1961) The recurrence theorem. Amer Math Mon 68:247–248

    MATH  Google Scholar 

  39. Young L-S (2002) What are SRB measures, and which dynamical systems have them? J Stat Phys 108:733–754

    MATH  Google Scholar 

Books and Reviews

  1. Billingsley P (1978) Ergodic Theory and Information. Robert E. Krieger Publishing Co, Huntington, pp xiii,194

    Google Scholar 

  2. Cornfeld IP, Fomin SV, Sinaĭ YG (1982) Ergodic Theory. Fundamental Principles of Mathematical Sciences, vol 245. Springer, New York, pp x,486

    Google Scholar 

  3. Denker M, Grillenberger C, Sigmund K (1976) Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics, vol 527. Springer, Berlin, pp iv,360

    Google Scholar 

  4. Friedman NA (1970) Introduction to Ergodic Theory. Van Nostrand Reinhold Co, New York, pp v,143

    Google Scholar 

  5. Glasner E (2003) Ergodic Theory via Joinings. Mathematical Surveys and Monographs, vol 101. American Mathematical Society, Providence, pp xii,384

    Google Scholar 

  6. Halmos PR (1960) Lectures on Ergodic Theory. Chelsea Publishing Co, New York, pp vii,101

    MATH  Google Scholar 

  7. 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, pp xviii,802

    Google Scholar 

  8. Hopf E (1937) Ergodentheorie. Ergebnisse der Mathematik und ihrer Grenzgebiete, 1st edn. Springer, Berlin, pp iv,83

    Google Scholar 

  9. Jacobs K (1960) Neue Methoden und Ereignisse der Ergodentheorie. Jber Dtsch Math Ver 67:143–182

    MathSciNet  Google Scholar 

  10. Petersen K (1989) Ergodic Theory. Cambridge Studies in Advanced Mathematics, vol 2. Cambridge University Press, Cambridge, pp xii,329

    Google Scholar 

  11. Royden HL (1988) Real Analysis, 3rd edn. Macmillan Publishing Company, New York, pp xx,444

    MATH  Google Scholar 

  12. Rudolph DJ (1990) Fundamentals of Measurable Dynamics. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, pp x,168

    Google Scholar 

  13. Walters P (1982) An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol 79. Springer, New York, pp ix,250

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of North Carolina, Chapel Hill, USA

    Karl Petersen

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  1. Karl Petersen
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Editors and Affiliations

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

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

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Petersen, K. (2009). Measure Preserving Systems. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_324

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