Journal of Statistical Physics

, Volume 113, Issue 1–2, pp 85–149 | Cite as

Random Versus Deterministic Exponents in a Rich Family of Diffeomorphisms

  • François Ledrappier
  • Michael Shub
  • Carles Simó
  • Amie Wilkinson


We study, both numerically and theoretically, the relationship between the random Lyapunov exponent of a family of area preserving diffeomorphisms of the 2-sphere and the mean of the Lyapunov exponents of the individual members. The motivation for this study is the hope that a rich enough family of diffeomorphisms will always have members with positive Lyapunov exponents, that is to say, positive entropy. At question is what sort of notion of richness would make such a conclusion valid. One type of richness of a family—invariance under the left action of SO(n+1)—occurs naturally in the context of volume preserving diffeomorphisms of the n-sphere. Based on some positive results for families linear maps obtained by Dedieu and Shub, we investigate the exponents of such a family on the 2-sphere. Again motivated by the linear case, we investigate whether there is in fact a lower bound for the mean of the Lyapunov exponents in terms of the random exponents (with respect to the push-forward of Haar measure on SO(3)) in such a family. The family ℱɛ that we study contains a twist map with stretching parameter ε. In the family ℱɛ, we find strong numerical evidence for the existence of such a lower bound on mean Lyapunov exponents, when the values of the stretching parameter ε are not too small. Even moderate values of ε like ε≥10 are enough to have an average of the metric entropy larger than that of the random map. For small ε the estimated average entropy seems positive but is definitely much less than the one of the random map. The numerical evidence is in favor of the existence of exponentially small lower and upper bounds (in the present example, with an analytic family). Finally, the effect of a small randomization of fixed size δ of the individual elements of the family ℱε is considered. Now the mean of the local random exponents of the family is indeed asymptotic to the random exponent of the entire family as ε tends to infinity.

Lyapunov estimates random diffeomorphism twist maps rich families 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Neishtadt, C. Simó, and A. Vasiliev, Geometric and statistical properties induced by separatrix crossings in volume-preserving systems, Nonlinearity 16:521-557 (2003).Google Scholar
  2. 2.
    K. Burns, C. Pugh, M. Shub, and A. Wilkinson, Recent results about stable ergodicity, Proc. Symposia A.M.S. 69:321-366 (2001).Google Scholar
  3. 3.
    A. Carverhill, Furstenberg's theorem for non-linear stochastic systems, Probab. Theory Related Fields 74:529-534 (1987).Google Scholar
  4. 4.
    J. P. Dedieu and M. Shub, On random and mean exponents for unitarily invariant probability measures on GL(n, C), to appear in Astérisque.Google Scholar
  5. 5.
    L. Carleson and T. Spencer, personal communicationGoogle Scholar
  6. 6.
    andT. Spencer, Standard Map Conjectures, Einstein Chair Lecture at CUNY, videotape #329.Google Scholar
  7. 7.
    Y. Kifer, Ergodic theory of random transformations, in Progress in Probability and Statistics, Vol. 10 (Birkhäuser Boston, Boston, MA, 1986).Google Scholar
  8. 8.
    Y. Kifer, Random perturbations of dynamical systems, in Progress in Probability and Statistics, Vol. 16 (Birkhäuser Boston, Boston, MA, 1988).Google Scholar
  9. 9.
    Y. Kifer, Random dynamics and its applications, in Proc. of Int. Congress of Math., Vol. II (Berlin, 1998). Doc. Math 1998, Extra Vol. II, 809-818 (electronic).Google Scholar
  10. 10.
    Pei-Dong Liu and Min Qian, Smooth ergodic theory of random dynamical systems, Lecture Notes in Math., No. 1606 (Springer, 1995).Google Scholar
  11. 11.
    I. Ya. Gol'shied and G. A. Margulis, Lyapunov indices of a product of random matrices, Russian Math. Surveys 44:11-71 (1989).Google Scholar
  12. 12.
    D. Ruelle, Ergodic Theory of Differentiable Dynamical Systems, Vol. 50 (Publications Mathématiques de l'IHES, 1979), pp. 27-58.Google Scholar
  13. 13.
    V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19:197-231 (1968).Google Scholar
  14. 14.
    A. Avila and J. Bochi, A formula with applications to the theory of Lyapunov exponents, Israel J. Math, to appear.Google Scholar
  15. 15.
    M. Do Carmo, Riemannian Geometry (Birkhauser, Boston, 1992).Google Scholar
  16. 16.
    A. Giorgilli, V. F. Lazutkin, and C. Simó, Visualization of a hyperbolic structure in area-preserving maps, Regular & Chaotic Dynamics 2:47-61 (1997).Google Scholar
  17. 17.
    H. Broer and C. Simó, Hill's equation with quasi-periodic forcing: Resonance tongues, instability pockets, and global phenomena, Bull. Soc. Bras. Mat. 29:253-293 (1998).Google Scholar
  18. 18.
    C. Simó and T. Stuchi, Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem, Physica D 140:1-32 (2000).Google Scholar
  19. 19.
    P. M. Cincotta and C. Simó, Simple tools to study global dynamics in non-axisymmetric galactic potentials, I, Astronom. & Astrophys. Supp. 147:205-228 (2000).Google Scholar
  20. 20.
    P. M. Cincotta, C. M. Giordano, and C. Simó, Phase space structure of multidimensional systems by means of the mean exponential growth factor of nearby orbits (MEGNO), Physica D, in press.Google Scholar
  21. 21.
    C. Simó, Global dynamics and fast indicators, in Global Analysis of Dynamical Systems, H. W. Broer et al., eds. (IOP Publishing, Bristol, 2001), pp. 373-390.Google Scholar
  22. 22.
    E. Fontich and C. Simó, The splitting of separatrices for analytic diffeomorphisms, Ergodic Theory Dynam. Systems 10:295-318 (1990).Google Scholar
  23. 23.
    A. Neishtadt, The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech. 48:133-139 (1984).Google Scholar
  24. 24.
    C. Simó, Invariant curves of perturbations of non twist integrable area preserving maps, Regular & Chaotic Dynamics 3:180-195 (1998).Google Scholar
  25. 25.
    H. Broer, R. Roussarie, and C. Simó, Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms, Ergodic Theory Dynam. Systems 16:1147-1172 (1996).Google Scholar
  26. 26.
    C. Simó, Averaging under fast quasi-periodic forcing, in Integrable and Chaotic Behaviour in Hamiltonian Systems, I. Seimenis, ed. (Plenum, New York, 1994), pp. 13-34.Google Scholar
  27. 27.
    C. Simó and C. Valls, A formal approximation of the splitting of separatrices in the classical Arnold's example of diffusion with two equal parameters, Nonlinearity 14:1707-1760 (2001).Google Scholar
  28. 28.
    R. Abraham and J. Robbin, Transversal Mappings and Flows (Benjamin, New York/Amsterdam, 1967).Google Scholar
  29. 29.
    R. Abraham and S. Smale, Nongenericity of Ω-stability, in 1970 Global Analysis, Proc. Sympos. Pure Math., Vol. 14 (Berkeley, CA, 1968), pp. 5-8.Google Scholar
  30. 30.
    D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Proc. Steklov. Inst. Math. 90(1967).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • François Ledrappier
    • 1
  • Michael Shub
    • 2
    • 3
  • Carles Simó
    • 4
  • Amie Wilkinson
    • 5
  1. 1.Centre de MathématiquesÉcole PolytechniquePalaiseau CedexFrance
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada
  3. 3.IBM T. J. Watson Research CenterYorktown Heights
  4. 4.Departamento de Matemàtica, Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain
  5. 5.Mathematics DepartmentNorthwestern UniversityEvanston

Personalised recommendations