Journal of Statistical Physics

, Volume 40, Issue 1–2, pp 69–91 | Cite as

On constructing Markov partitions by computer

  • Valter Franceschini
  • Fernando Zironi


Two methods to construct Markov partitions for two-dimensional systems are proposed. One is based on the existence of a known, or easily accessible by numerical analysis, hyperbolic fixed point; the other one, which is more general, is derived from Bowen's proof of the existence theorem of Markov partitions for hyperbolic systems. The methods are successfully implemented in two cases of hyperbolic systems: the linear automorphism\(\left( {\begin{array}{*{20}c} 1 & 1 \\ 1 & 2 \\ \end{array} } \right)\) of the 2-torus and a nonlinear perturbation of it. The methods are applied also to the Hénon mapping. In such nonhyperbolic case, however, they produce partitions of the Hénon attractor which lack some essential properties.

Key words

Symbolic dynamics Markov partitions hyperbolic systems fixed point stable and unstable manifolds Hénon map 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. M. Alekseev and M. V. Yacobson, Symbolic dynamics and hyperbolic systems,Phys. Rep. 75:287–325 (1981).Google Scholar
  2. 2.
    G. Gallavotti,Aspetti della teoria ergodica, qualitativa e statistica del moto, Quaderni dell'UMI 21 (1981).Google Scholar
  3. 3.
    R. Bowen, Markov partitions for AxiomA diffeomorphisms,Am. J. Math. 92:725–747 (1970).Google Scholar
  4. 4.
    Ya. G. Sinai, Markov partitions andC-diffeomorphisms,Funct. Anal. Appl. 2:61–82 (1968).Google Scholar
  5. 5.
    Ya. G. Sinai, Construction of Markov partitions,Funct. Anal. Appl. 2:245–253 (1968).Google Scholar
  6. 6.
    R. L. Adler and B. Weiss, Similarity of automorphisms of the torus,Mem. Am. Math. Soc. 98 (1970).Google Scholar
  7. 7.
    Ya. G. Sinai, Gibbs measures in ergodic theory,Russ. Math. Surv. 166:21–69 (1972).Google Scholar
  8. 8.
    M. Hénon, A two-dimensional mapping with a strange attractor,Commun. Math. Phys. 50:69–77 (1976).Google Scholar
  9. 9.
    P. Collet and Y. Levy, Ergodic properties of the Lozi mappings,Commun. Math. Phys., 461–481 (1984).Google Scholar
  10. 10.
    D. Ruelle, Thermodinamic formalism,Encyclopedia of Mathematics (Addison-Wesley, Reading, Massachusetts, 1978), pp. 125–130.Google Scholar
  11. 11.
    V. I. Arnold and A. Avez,Problèmes ergodiques de la mecanique classique (Gauthier-Villars, Paris, 1967).Google Scholar
  12. 12.
    R. L. Adler and B. Weiss, Entropy, a complete metric invariant for automorphisms of the torus,Proc. Natl. Acad. Sci. USA 57:1573–1576 (1967).Google Scholar
  13. 13.
    V. Franceschini and L. Russo, Stable and unstable manifolds of the Hénon mapping,J. Stat. Phys. 25:757–769 (1981).Google Scholar

Copyright information

© Plenum Publishing Corporation 1985

Authors and Affiliations

  • Valter Franceschini
    • 1
  • Fernando Zironi
    • 1
  1. 1.Dipartimento di Matematica Pura ed ApplicataUniversita' di ModenaModenaItaly

Personalised recommendations