Journal of Statistical Physics

, Volume 50, Issue 1–2, pp 213–229 | Cite as

A matrix method for estimating the Liapunov exponent of one-dimensional systems

  • Abraham Boyarsky


Let τ: [0, 1]→[0, 1] be a piecewise monotonie expanding map. Then τ admits an absolutely continuous invariant measureμ. A result of Kosyakin and Sandler shows thatμ can be approximated by a sequence of absolutely continuous measuresμ n invariant under piecewise linear Markov maps τitn. Each τitn is constructed on the inverse images of the turning points of τ. The easily computable measuresμ n are used to estimate the Liapunov exponent of τ. The idea of using Markov maps for estimating the Liapunov exponent is applied to both expanding and nonexpanding maps.

Key words

Liapunov exponent piecewise monotonie map Markov map absolutely continuous univariant measure negative Schawarzian 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Wolf and J. Swift, inStatistical Physics and Chaos in Fusion Plasmas, C. W. Horton, Jr., and L. E. Reichl, eds. (Wiley, 1984).Google Scholar
  2. 2.
    A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano,Physica 16D:285–317 (1985).Google Scholar
  3. 3.
    J.-C. Roux, R. H. Simoyi, and H. L. Swinney,Physica 8D:257–266 (1983).Google Scholar
  4. 4.
    A. Wolf, inChaos, A. V. Holden, zs. (Manchester University Press, 1986).Google Scholar
  5. 5.
    A. Wolf and J. A. Vastano, inDimension and Entropies in Chaotic Systems, G. Mayer-Kress, ed. (Springer-Verlag, 1987).Google Scholar
  6. 6.
    J. Wright,Phys. Rev. A 29(5):2924–2927 (1984).Google Scholar
  7. 7.
    L.-S. Young,IEEE Trans. Circuits Systems CAS-30(8):599–607 (1983).Google Scholar
  8. 8.
    F. Ledrappier,Ergodic. Theory Dynam. Syst. 1:77–93 (1981).Google Scholar
  9. 9.
    A. A. Kosyakin and E. A. Sandler,Izv. VUZ Matemat. (3) [118], 32–40 (1972). (English translation available from the British Library, Translation Service.)Google Scholar
  10. 10.
    M. Sano and Y. Sawada,Phys. Rev. Lett. 55(10):1082–1085 (1985).Google Scholar
  11. 11.
    A. Boyarsky,J. Math. Anal. Appl. 76:483–497 (1980).Google Scholar
  12. 12.
    G. Choquet,Lectures on Analysis, Vol. I (Benjamin, New York, 1969).Google Scholar
  13. 13.
    A. Katok and Y. Kifer,Tour. Anal. Math. 47:193–237 (1986).Google Scholar
  14. 14.
    A. Boyarsky, Singular perturbations of piecewise monotonie maps of the interval, preprint.Google Scholar
  15. 15.
    H. Proppe, W. Byers, and A. Boyarsky,Isr. J. Math. 44(4):277–288 (1983).Google Scholar
  16. 16.
    V. I. Oseledec,Trans. Moscow Math. Soc. 19:197–231 (1968).Google Scholar
  17. 17.
    A. M. Liapunov,Ann. Math. Study 17 (1947).Google Scholar
  18. 18.
    G. Bennetin, L. Galgani, and J. M. Strelcyn,Meccanica 15:9–20 (1980).Google Scholar
  19. 19.
    H. A. Lauwener, inChaos, A. V. Holden, ed. (Princeton University Press, 1986).Google Scholar
  20. 20.
    M. Denker, C. Gillenberger, and K. Sigmund,Ergodic Theory on Compact Spaces (Springer-Verlag, New York, 1976).Google Scholar
  21. 21.
    A. Lasota and J. A. Yorke,Trans. Am. Math. Soc. 186:481–488 (1973).Google Scholar
  22. 22.
    A. Boyarsky and M. Scarowsky,Trans. Am. Math. Soc. 255:243–262 (1979).Google Scholar
  23. 23.
    M. Misiurewicz,Inst. Hautes Etudes Sci. Publ. Math. 53 (1981).Google Scholar
  24. 24.
    M. Misiurewicz,Inst. Hautes Etudes Sci. Publ. Math. 51 (1979).Google Scholar
  25. 25.
    R. Bowen,Commun. Math. Phys. 69:1–17 (1979).Google Scholar
  26. 26.
    S. D. Johnson, Continuous measures and strange attractors in one dimension, Ph. D. Dissertation, Stanford University (1985).Google Scholar
  27. 27.
    N. Friedman and A. Boyarsky,Lin. Alg. Appl. 38:141–147 (1981).Google Scholar
  28. 28.
    G. Choquet,Lectures on Analysis, Vol. I (Benjamin, New York, 1969).Google Scholar
  29. 29.
    F. Topsoe,Ann. Math. Stat. 38:1661–1665 (1967).Google Scholar
  30. 30.
    B. L. S. Prakasa Rao,Non Parametric Functional Estimation (Academic Press, 1983).Google Scholar
  31. 31.
    L. Arnold and V. Wihstutz, Liapunov exponents: A survey, Report No. 141, University of Bremen (1985).Google Scholar
  32. 32.
    J. Guckenheimer,Commun. Math. Phys. 70:133–160 (1979).Google Scholar
  33. 33.
    L. Arnold, Stochastic systems: Qualitative theory and Liapunov exponents, preprint.Google Scholar
  34. 34.
    P. Góra,Coll. Math. XLIL:73–85 (1984).Google Scholar
  35. 35.
    P. Góra, A. Boyarsky, H. Proppe, Constructive approximations to densities invariant under non-expanding transformations,Jour. Stat. Physics, in press.Google Scholar

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • Abraham Boyarsky
    • 1
  1. 1.Department of Mathematics, Loyola CampusConcordia UniversityMontrealCanada

Personalised recommendations