Ergodic Theory of Nonlinear Dynamics

  • Alfredo Medio
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 476)


This lectures provides an introduction to ergodic theory — the study of probabilistic properties of orbits generated by continuous- or discrete-time dynamical systems. It includes some elementary measure theory, together with the basic concepts and methods of ergodic theory, such as invariant, ergodic, natural probability measures, isomorphism and metric entropy. The notion of predictability of dynamical systems and the related definition of complex or chaotic dynamics are discussed and some basic results are applied to certain types of maps commonly used in applications. Finally, the lecture discusses Bernoulli systems, the relation between deterministic chaotic systems and stochastic processes, and the notion of α-congruence.


Invariant Measure Ergodic Theory Measurable Subset Topological Entropy Ergodic Measure 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Billingsley, P. 1965. Ergodic Theory and Information. New York: John Wiley.MATHGoogle Scholar
  2. Billingsley, P. 1979. Probability and Measure. New York: John Wiley.MATHGoogle Scholar
  3. Cornfeld, I. P., Fomin, S. V. and Sinai, Ya. G. 1982. Ergodic Theory. New York: Springer-Verlag.MATHGoogle Scholar
  4. Cornfeld, I. P. and Sinai, Ya. G. 2000. General ergodic theory of groups of measure preserving transformations, in Ya.G. Sinai (ed.), Dynamical Sty stems, Ergodic Theory and Applications. Berlin: Springer-Verlag.Google Scholar
  5. Doob, J. L. 1994. Measure Theory. New York: Springer-Verlag.MATHGoogle Scholar
  6. Eckmann, J. P. and Ruelle, D. 1985. Ergodic theory of chaos and strange attractors. Reviews of Modern Physics, 57:617–56.CrossRefADSMathSciNetGoogle Scholar
  7. Jakobson, M. V. 2000. Ergodic theory of one-dimensional mappings, in Ya. G. Sinai (ed.), Dynamical Stystems, Ergodic Theory and Applications. Berlin: Springer-Verlag.Google Scholar
  8. Katok, A. and Hasselblatt, B. 1995. Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press.MATHGoogle Scholar
  9. Keller, G. 1998. Equilibrium States in Ergodic Theory. Cambridge: Cambridge University Press.MATHGoogle Scholar
  10. MacKay, R. S. 1992. Nonlinear dynmaics in economics: a review of some key features of nonlinear dynamical systems. Florence: European University Institute, manuscript.Google Scholar
  11. Mañe, R. 1987. Ergodic Theory and Differentiable Dynamics. Berlin: Springer-Verlag.MATHGoogle Scholar
  12. Oono, Y. and Osikawa, M. 1980. Chaos in nonlinear difference equations, I. Progress of Theoretical Physics, 64:54–67.MATHCrossRefADSMathSciNetGoogle Scholar
  13. Ornstein, D. S. 1974. Ergodic Theory, Randomness, and Dynamical Systems, Yale Mathematical Monographs 5. New Haven and London: Yale University Press.MATHGoogle Scholar
  14. Ornstein, D. S. and Weiss, B. 1991. Statistical properties of chaotic systems. Bulletin (New Series) of the American Mathematical Society, 24:11–115.MATHMathSciNetCrossRefGoogle Scholar
  15. Osledec, V. I. 1968. A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems. Transactions of the Moscow Mathematical Society, 19:197–221.Google Scholar
  16. Radunskaya, A. 1992. Alpha-Congruence of Bernoulli Flows and Markov Processes: Distinguishing Random and Deterministic Chaos, PhD thesis. Palo Alto: Stanford University, unpublished.Google Scholar
  17. Ruelle, D. 1989. Chaotic Evolution and Strange Attractors. Cambridge: Cambridge University Press.MATHCrossRefGoogle Scholar
  18. Walters, P. 1982. An Introduction of Ergodic Theory. Berlin: Springer-Verlag.Google Scholar

Copyright information

© CISM, Udine 2005

Authors and Affiliations

  • Alfredo Medio
    • 1
  1. 1.Department of StatisticsUniversity of UdineUdineItaly

Personalised recommendations