Monatshefte für Mathematik

, Volume 94, Issue 4, pp 313–333

Stochastic stability in some chaotic dynamical systems

  • Gerhard Keller
Article

Abstract

For a certain class of piecewise monotonic transformations it is shown using a spectral decomposition of the Perron-Frobenius-operator ofT that invariant measures depend continuously on 3 types of perturbations: 1) deterministic perturbations, 2) stochastic perturbations, 3) randomly occuring deterministic perturbations. The topology on the space of perturbed transformations is derived from a metric on the space of Perron-Frobenius-operators.

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References

  1. [1]
    Góra, P.: On small stochastic perturbations of one-sided subshifts of finite type. Bull. Acad. Pol. Sc.27, 47–51 (1979).Google Scholar
  2. [2]
    Góra, P.: On small stochastic perturbations of mappings on the unit interval. Preprint. Inst. Math., Warsaw Univ.Google Scholar
  3. [3]
    Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z.180, 119–140 (1982).Google Scholar
  4. [4]
    Hofbauer, F., Keller, G.: Equilibrium states for piecewise monotonic transformations. Preprint. Univ. Heidelberg.Google Scholar
  5. [5]
    Ionescu Tulcea, C. T., Marinescu, G.: Théorie ergodique pour des classes d'opérations non complètement continues. Ann. Math.52, 140–147 (1950).Google Scholar
  6. [6]
    Keller, G.: Un théorème de la limite centrale pour une classe de transformations monotones par morceaux. C. R. Acad. Sci. Paris, Série A291, 155–158 (1980).Google Scholar
  7. [7]
    Keller, G.: Ergodicité et mesure invariantes pour les transformations dilatantes par morceaux d'une région bornée du plan. C. R. Acad. Sci. Paris, Série A289, 625–627 (1979).Google Scholar
  8. [8]
    Keller, G.: Propriétés ergodiques des endomorphismes dilatants,C 2 par morceaux, des régions bornées du plan. Thèse 3e cycle. Université de Rennes. 1979.Google Scholar
  9. [9]
    Kowalski, Z.: Invariant measures for piecewise monotonic transformations. In: Proc. 4th Winter-School on Prob., Karpacz, Poland, 1975, pp. 77–94. Lect. Notes Math. 472. Berlin-Heidelberg-New York: Springer, 1975.Google Scholar
  10. [10]
    Lasota, A., Yorke, J. A.: On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc.186, 481–488 (1973).Google Scholar
  11. [11]
    Li, T. Y.: Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture.J. Approx. Th.17, 177–186 (1976).Google Scholar
  12. [12]
    Schaefer, H. H.: Banach Lattices and Positive Operators. Berlin-Heidelberg-New York: Springer. 1974.Google Scholar
  13. [13]
    Schaefer, H. H.: On positive contractions inL p-spaces. Trans. Amer. Math. Soc.257, 261–268 (1980).Google Scholar
  14. [14]
    Wong, S.: Some metric properties of piecewise monotonic mappings of the unit interval. Trans. Amer. Math. Soc.246, 493–500 (1978).Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Gerhard Keller
    • 1
  1. 1.Institut für Angewandte MathematikUniversität HeidelbergHeidelbergGermany

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