Communications in Mathematical Physics

, Volume 156, Issue 2, pp 355–385 | Cite as

On the spectra of randomly perturbed expanding maps

  • V. Baladi
  • L. -S. Young


We consider small random perturbations of expanding and piecewise expanding maps and prove the robustness of their invariant densities and rates of mixing. We do this by proving the robustness of the spectra of their Perron-Frobenius operators.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baladi, V., Keller, G.: Zeta functions and transfer operators for piecewise monotone transformations. Commun. Math. Phys.127, 459–477 (1990)Google Scholar
  2. 2.
    Baxendale, P.: Brownian motions in the diffeomorphism group. Compositio Math.53, 19–50 (1984)Google Scholar
  3. 3.
    Benedicks, M., Young, L.-S.: Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps. Ergodic Theory Dynamical Systems12, 13–37 (1992)Google Scholar
  4. 4.
    Blank, M.L.: Chaotic mappings and stochastic Markov chains. Mathematical Physics X. Proceedings of the Xth Congress on Mathematical Physics, Leipzig, Germany, 1991, Schmüdgen, K. (ed.) Berlin, Heidelberg, New York: Springer 1992, pp. 341–345Google Scholar
  5. 5.
    Collet, P.: Ergodic properties of some unimodal mappings of the interval. Preprint Mittag-Leffler (1984)Google Scholar
  6. 6.
    Collet, P.: Some ergodic properties of maps of the interval. Dynamical Systems and Frustrated Systems. Bamon, R., Gambaudo, J.-M., Martinez, S. (eds.) 1991 (to appear)Google Scholar
  7. 7.
    Collet, P., Isola, S.: On the essential spectrum of the transfer operator for expanding Markov maps. Commun. Math. Phys.139, 551–557 (1991)Google Scholar
  8. 8.
    Coven, E.M., Kan, I., Yorke, J.A.: Pseudo-orbit shadowing in the family of tent maps. Trans. Am. Math. Soc.308, 227–241 (1988)Google Scholar
  9. 9.
    Franks, J.: Manifolds ofC r mappings and applications to differentiable dynamical systems. Studies in Analysis, Adv. Math. Suppl. Stud.4, 271–291 (1979)Google Scholar
  10. 10.
    Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z.180, 119–140 (1982)Google Scholar
  11. 11.
    Keller, G.: Stochastic stability in some chaotic dynamical systems. Monatsh. Math.94, 313–333 (1982)Google Scholar
  12. 12.
    Keller, G.: On the rate of convergence to equilibrium in one-dimensional systems. Commun. Math. Phys.96, 181–193 (1984)Google Scholar
  13. 13.
    Kifer, Y.: On small random perturbations of some smooth dynamical systems. Math. USSR-Izv.8, 1083–1107 (1974)Google Scholar
  14. 14.
    Kifer, Y.: Ergodic theory of random transformations. Boston, Basel: Birkhäuser 1986Google Scholar
  15. 15.
    Kifer, Y.: Random perturbations of dynamical systems. Boston, Basel: Birkhäuser 1988aGoogle Scholar
  16. 16.
    Kifer, Y.: A note on integrability ofC r norms of stochastic flows and applications. Stochastic Mechanics and Stochastic Processes, Proc. Conf. Swansea/UK 1986, Lecture Notes in Math.1325, Berlin, Heidelberg, New York: Springer 1988b, pp. 125–131Google Scholar
  17. 17.
    Kunita, H.: Stochastic flows and stochastic differential equations. Cambridge: Cambridge University Press 1990Google Scholar
  18. 18.
    Mañé, R.: Ergodic theory and differentiable dynamics. Berlin, Heidelberg, New York: Springer 1987Google Scholar
  19. 19.
    Mayer, D.: On a ξ function related to the continued fraction transformation. Bull. Soc. Math. France104, 195–203 (1976)Google Scholar
  20. 20.
    Ruelle, D.: Zeta functions for expanding maps and Anosov flows. Invent. Math.34, 231–242 (1976)Google Scholar
  21. 21.
    Ruelle, D.: Locating resonances for Axiom A dynamical systems. J. Stat. Phys.44, 281–292 (1986)Google Scholar
  22. 22.
    Ruelle, D.: The thermodynamic formalism for expanding maps. Commun. Math. Phys.125, 239–262 (1989)Google Scholar
  23. 23.
    Ruelle, D.: An extension of the theory of Fredholm determinants. Inst. Hautes Études Sci. Publ. Math.72, 175–193 (1990)Google Scholar
  24. 24.
    Rychlik, M.: Bounded variation and invariant measures. Studia Math.LXXVI, 69–80 (1983)Google Scholar
  25. 25.
    Wilkinson, J.H.: The Algebraic Eigenvalue Problem. London: Oxford University Press 1965Google Scholar
  26. 26.
    Wong, S.: Some metric properties of piecewise monotonic mappings of the unit interval. Trans. Am. Math. Soc.246, 493–500 (1978)Google Scholar
  27. 27.
    Yosida, K.: Functional Analysis (Sixth Edition). deGrundlehren, der mathematischen Wissenschaften123. Berlin, Heidelberg, New York: Springer 1980Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • V. Baladi
    • 1
  • L. -S. Young
    • 2
    • 3
  1. 1.UMR 128, UMPA, ENS Lyon, 46CNRSLyon CedexFrance
  2. 2.Department of MathematicsUniversity of ArizonaTucsonUSA
  3. 3.Department of MathematicsUCLALos AngelesUSA

Personalised recommendations