Skip to main content
SpringerLink
Log in

Ergodic Theory

  • Chapter
Time Series and Statistics

Part of the book series: The New Palgrave ((NPA))

Abstract

To begin in the middle; for that is where ergodic theory started, in the middle of the development of statistical mechanics, with the solution, by von Neumann and Birkhoff, of the problem of identifying space averages with time averages. This problem can be formulated as follows: If xl(− ∞ < t < ∞) represents the trajectory (orbit) passing through the point x = x0 at time t = 0 of a conservative dynamical system, when can one make the identification

$$\left( * \right)\mathop {\lim }\limits_{T \to \infty } \left( {1/T} \right)\int_0^T {f\left( {{x_t}} \right){\text{d}}t} = \int_\Omega {f{\text{d}}m/m\left( \Omega \right)}$$

for suitable functions defined on the phase space Ω of the system?

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 29.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 39.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  • Abramov, L.M. 1962. Metric automorphisms with quasi-discrete spectrum. Izvestiya Akademii Nauk Ser. Mat. 26, 513–30; American Mathematical Society Translations 2(39), 37–56.

    Google Scholar 

  • Adler, R.L. and Marcus, B. 1979. Topological entropy and equivalence of dynamical systems. Memoirs of the American Mathematical Society 219, 1–84.

    Google Scholar 

  • Anosov, D.V. 1967. Geodesic flows on closed Riemannian manifolds with negative curvature. Trudy. Mat. Inst. Steklova 90, 1–209; Proceedings of the Steklov Institute of Mathematics (American Mathematical Society Translations), 1969, 1–235.

    Google Scholar 

  • Auslander, L., Green, L. and Hahn, F. 1963. Flows on homogeneous spaces. Annals of Mathematics Studies 53, Princeton.

    Google Scholar 

  • Birkhoff, G.D. 1931. Proof of the ergodic theorem. Proceedings of the National Academy of Sciences of the USA 17, 656–60.

    Article  Google Scholar 

  • Bowen, R. 1977. On Axiom A diffeomorphisms. American Mathematical Society Regional Conference Series 35, 1–45.

    Google Scholar 

  • Chacon, R.V. and Ornstein, D.S. 1960, A general ergodic theorem. Illinois Journal of Mathematics 4, 153–60.

    Google Scholar 

  • Collet, P. and Eckmann, J.P. 1980. Iterated Maps on the Interval as Dynamical Systems. Progress in Physics, Vol. 1, Boston: Birkhauser.

    Google Scholar 

  • Feldman, J. 1976. Non-Bernoulli K-automorphisms and a problem of Kakutani. Israel Journal of Mathematics 24, 16–37.

    Article  Google Scholar 

  • Furstenberg, H. 1961. Strict ergodicity and transformations of the torus. American Journal of Mathematics 83, 573–601.

    Article  Google Scholar 

  • Furstenberg, H. 1977. Ergodic behaviour of diagonal measures and a theorem of Szemeredi on arithmetic progressions. Journal d’analyse mathématique 31, 2204–56.

    Article  Google Scholar 

  • Halmos, P.R. and von Neumann, J. 1942. Operator methods in classical mechanics II. Annals of Mathematics 43, 332–50.

    Article  Google Scholar 

  • Hejhal, D.A. 1976. The Selberg trace formula and the Riemann zeta function. Duke Mathematical Journal 43, 441–82.

    Article  Google Scholar 

  • Kakutani, S. 1943. Induced measure-preserving transformations. Proceedings of the Imperial Academy of Tokyo 19, 635–41.

    Article  Google Scholar 

  • Katok, A. 1977. Monotone equivalence in ergodic theory. Izvestiya Akademii Nauk Ser. Mat. 41, 104–157.

    Google Scholar 

  • Kolmogorov, A.N. 1958. A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces. Doklady Akademii Nauk SSSR 119, 8561–864 (Russian).

    Google Scholar 

  • Ornstein, D.S. 1970. Bernoulli shifts with the same entropy are isomorphic. Advances in Mathematics 4, 337–52.

    Article  Google Scholar 

  • Ornstein, D.S. and Weiss 1984. Any flow is the orbit factor of any other. Ergodic Theory and Dynamical Systems 4, 105–16.

    Article  Google Scholar 

  • Parry, W. 1971. Metric classifications of ergodic nil flows and unipotent affines. American Journal of Mathematics 93, 819–28.

    Article  Google Scholar 

  • Parry, W. and Pollicott, M. 1983. An analogue of the prime number theorem for closed orbits of Axiom A flows. Annals of Mathematics 118, 573–91.

    Article  Google Scholar 

  • Patterson, S.J. 1976. The limit set of a Fuchsian group. Acta Mathematica 136, 241–73.

    Article  Google Scholar 

  • Pesin, J. 1977. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Mathematical Surveys 32(4), 55–114.

    Article  Google Scholar 

  • Ratner, M. 1982. Rigidity of horocycle flows. Annals of Mathematics 115, 597–614.

    Article  Google Scholar 

  • Rees, M. 1982. Positive measure sets of ergodic rational maps. University of Minnesota Mathematics Report.

    Google Scholar 

  • Rudolph, D. 1984. Restricted Orbit Equivalence. Reprinted, Baltimore University of Maryland.

    Google Scholar 

  • Ruelle, D. 1978. Thermodynamic Formalism. Reading, Mass.: Addison-Wesley.

    Google Scholar 

  • Ruelle, D. and Takens, F. 1971. On the nature of turbulence. Communications in Mathematical Physics 20, 167–92.

    Article  Google Scholar 

  • Schmidt, K. 1977. Cocycles on Ergodic Transformation Groups. London: Macmillan.

    Google Scholar 

  • Sinai, J.G. 1959. On the concept of entropy of a dynamical system. Doklady Akademii Nauk SSSR 124, 768–71.

    Google Scholar 

  • Sinai, J.G. 1963. On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics. Doklady Akademii SSSR 153, 1261–4; Sov. Math. Dokl. 4 (1963), 1818–22.

    Google Scholar 

  • Sinai, J.G. 1972. Gibbsian measures in ergodic theory. Usphehi Matematiceskich Nauk 27, No. 4, 21–64; Russian Mathematical Surveys 27(4), 21–69.

    Google Scholar 

  • Smale, S. 1967. Differentiable dynamical systems. Bulletin of the American Mathematical Society 73, 747–817.

    Article  Google Scholar 

  • Sullivan, D. 1979. The density at infinity of a discrete group of hyperbolic motions. Publications mathematiques 50, 419–450.

    Google Scholar 

  • Von Neumann, J. 1932a. Proof of the quasi-ergodic hypothesis. Proceedings of the National Academy of Sciences of the USA 18, 70–82.

    Article  Google Scholar 

  • Von Neumann, J. 1932b. Zur operatoren Methode in der klassischen Mechanik. Annals of Mathematics 33, 587–642.

    Article  Google Scholar 

  • Walters, P. 1973. A variational principle for the pressure of continuous transformations. American Journal of Mathematics 97, 937–71.

    Article  Google Scholar 

  • Williams, R.F. 1973. Classification of subshifts of finite type. Annals of Mathematics 88, 120–93.

    Article  Google Scholar 

Download references

Authors
  1. William Parry
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

John Eatwell Murray Milgate Peter Newman

Copyright information

© 1990 Palgrave Macmillan, a division of Macmillan Publishers Limited

About this chapter

Cite this chapter

Parry, W. (1990). Ergodic Theory. In: Eatwell, J., Milgate, M., Newman, P. (eds) Time Series and Statistics. The New Palgrave. Palgrave Macmillan, London. https://doi.org/10.1007/978-1-349-20865-4_7

Download citation

  • DOI: https://doi.org/10.1007/978-1-349-20865-4_7

  • Publisher Name: Palgrave Macmillan, London

  • Print ISBN: 978-0-333-49551-3

  • Online ISBN: 978-1-349-20865-4

  • eBook Packages: Palgrave History CollectionHistory (R0)

Publish with us

Policies and ethics

Search

Navigation