The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Ergodic Theory

  • William Parry
Reference work entry


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 xI(− ∞ < 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
$$ \left({}^{\ast}\right) \lim \limits_{T\to \infty } \left(1/T\right){\int}_0^Tf\left({x}_t\right)\mathrm{d}t={\int}_{\Omega}f\;\mathrm{d}m/m\left(\Omega \right) $$
for suitable functions defined on the phase space Ω of the system?
This is a preview of subscription content, log in to check access.


  1. Abramov, L.M. 1962. Metric automorphisms with quasi-discrete spectrum. Izvestiya Akademii Nauk Seriya Matematicheskaya 26: 513–530; American Mathematical Society Translations 2(39): 37–56.Google Scholar
  2. Adler, R.L., and B. Marcus. 1979. Topological entropy and equivalence of dynamical systems. Memoirs of the American Mathematical Society 219: 1–84.Google Scholar
  3. Anosov, D.V. 1967. Geodesic flows on closed Riemannian manifolds with negative curvature. Trudy Matematicheskogo Instituta imeni VA Steklova 90: 1–209; Proceedings of the Steklov Institute of Mathematics (American Mathematical Society Translations), 1969, 1–235.Google Scholar
  4. Auslander, L., L. Green, and F. Hahn. 1963. Flows on homogeneous spaces, Annals of mathematics studies, vol. 53. Princeton: Princeton University Press.Google Scholar
  5. Birkhoff, G.D. 1931. Proof of the ergodic theorem. Proceedings of the National Academy of Sciences of the United States of America 17: 656–660.CrossRefGoogle Scholar
  6. Bowen, R. 1977. On axiom A diffeomorphisms. American Mathematical Society Regional Conference Series 35: 1–45.Google Scholar
  7. Chacon, R.V., and D.S. Ornstein. 1960. A general ergodic theorem. Illinois Journal of Mathematics 4: 153–160.Google Scholar
  8. Collet, P., and J.P. Eckmann. 1980. Iterated maps on the interval as dynamical systems. Progress in physics, vol. 1. Boston: Birkhauser.Google Scholar
  9. Feldman, J. 1976. Non-Bernoulli K-automorphisms and a problem of Kakutani. Israel Journal of Mathematics 24: 16–37.CrossRefGoogle Scholar
  10. Furstenberg, H. 1961. Strict ergodicity and transformations of the torus. American Journal of Mathematics 83: 573–601.CrossRefGoogle Scholar
  11. Furstenberg, H. 1977. Ergodic behaviour of diagonal measures and a theorem of Szemeredi on arithmetic progressions. Journal d’analyse mathématique 31: 2204–2256.CrossRefGoogle Scholar
  12. Halmos, P.R., and J. von Neumann. 1942. Operator methods in classical mechanics II. Annals of Mathematics 43: 332–350.CrossRefGoogle Scholar
  13. Hejhal, D.A. 1976. The Selberg trace formula and the Riemann zeta function. Duke Mathematical Journal 43: 441–482.CrossRefGoogle Scholar
  14. Kakutani, S. 1943. Induced measure-preserving transformations. Proceedings of the Imperial Academy of Tokyo 19: 635–641.CrossRefGoogle Scholar
  15. Katok, A. 1977. Monotone equivalence in ergodic theory. Izvestiya Akademii Nauk Seriya Matematicheskaya 41: 104–157.Google Scholar
  16. Kolmogorov, A.N. 1958. A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces. Doklady Akademii Nauk SSSR 119: 8561–8864 (Russian).Google Scholar
  17. Ornstein, D.S. 1970. Bernoulli shifts with the same entropy are isomorphic. Advances in Mathematics 4: 337–352.CrossRefGoogle Scholar
  18. Ornstein, D.S., and B. Weiss. 1984. Any flow is the orbit factor of any other. Ergodic Theory and Dynamical Systems 4: 105–116.CrossRefGoogle Scholar
  19. Parry, W. 1971. Metric classifications of ergodic nil flows and unipotent affines. American Journal of Mathematics 93: 819–828.CrossRefGoogle Scholar
  20. Parry, W., and M. Pollicott. 1983. An analogue of the prime number theorem for closed orbits of axiom A flows. Annals of Mathematics 118: 573–591.CrossRefGoogle Scholar
  21. Patterson, S.J. 1976. The limit set of a Fuchsian group. Acta Math 136: 241–273.CrossRefGoogle Scholar
  22. Pesin, J. 1977. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Mathematical Surveys 32(4): 55–114.CrossRefGoogle Scholar
  23. Ratner, M. 1982. Rigidity of horocycle flows. Annals of Mathematics 115: 597–614.CrossRefGoogle Scholar
  24. Rees, M. 1982. Positive measure sets of ergodic rational maps. University of Minnesota Mathematics report.Google Scholar
  25. Rudolph, D. 1984. Restricted orbit equivalence. Reprinted, Baltimore University of Maryland.Google Scholar
  26. Ruelle, D. 1978. Thermodynamic formalism. Reading: Addison-Wesley.Google Scholar
  27. Ruelle, D., and F. Takens. 1971. On the nature of turbulence. Communications in Mathematical Physics 20: 167–192.CrossRefGoogle Scholar
  28. Schmidt, K. 1977. Cocycles on ergodic transformation groups. London: Macmillan.Google Scholar
  29. Sinai, J.G. 1959. On the concept of entropy of a dynamical system. Doklady Akademii Nauk SSSR 124: 768–771.Google Scholar
  30. Sinai, J.G. 1963. On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics. Doklady Akademii Nauk SSSR 153: 1261–1264; Soviet Mathematics - Doklady 4: 1818–1822, 1963.Google Scholar
  31. Sinai, J.G. 1972. Gibbsian measures in ergodic theory. Usphehi Matematiceskich Nauk 27(4): 21–64; Russian Mathematical Surveys 27(4): 21–69.Google Scholar
  32. Smale, S. 1967. Differentiable dynamical systems. Bulletin of the American Mathematical Society 73: 747–817.CrossRefGoogle Scholar
  33. Sullivan, D. 1979. The density at infinity of a discrete group of hyperbolic motions. Publications mathématiques 50: 419–450.CrossRefGoogle Scholar
  34. Von Neumann, J. 1932a. Proof of the quasi-ergodic hypothesis. Proceedings of the National Academy of Sciences of the United States of America 18: 70–82.CrossRefGoogle Scholar
  35. Von Neumann, J. 1932b. Zur operatoren Methode in der klassischen Mechanik. Annals of Mathematics 33: 587–642.CrossRefGoogle Scholar
  36. Walters, P. 1973. A variational principle for the pressure of continuous transformations. American Journal of Mathematics 97: 937–971.CrossRefGoogle Scholar
  37. Williams, R.F. 1973. Classification of subshifts of finite type. Annals of Mathematics 88: 120–193.CrossRefGoogle Scholar

Copyright information

© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • William Parry
    • 1
  1. 1.