Random composing of mappings, small stochastic perturbations and attractors

  • Paweł Góra


Let T and H r (0<rr0) be continuous mappings of a compact metric space (M,d), d](x,Hr(x))≦r for any x∈M. We consider Markov processes \(\tilde T_r \) with transition functions
$$\tilde P^r (x,A) = p\chi _A (T(x)) + q\chi _A (H_r (x))$$
. They are random compositions of T and H r . We study the existence, uniqueness and asymptotic (r→0) behaviour of \(\tilde T_r \)-invariant measures \(\tilde \mu _r \). We do this by converting the problem into the problem of small stochastic perturbations of the mapping T. The main result is that the weak limit points (for r→0) of the set \(\{ \tilde \mu _r :{\text{ 0 < }}r \leqq r_0 \} \) are measures concentrated on “attractors” of the mapping T.

Our definition of “attractors” is based on ideas similar to those of Ruelle [5].

The perturbations we deal with are nonlocal and singular, whereas so far most authors considered local and “absolutely continuous” stochastic perturbations (e.g. Ruelle [5]).


Continuous Mapping Stochastic Process Probability Theory Markov Process Invariant Measure 
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.
    Doob, J.L.: Stochastic Processes. New York: J. Wiley 1953Google Scholar
  2. 2.
    Furstenberg, H.: Noncommuting random products. Trans. Amer. Math. Soc. 108, 377–428 (1963)Google Scholar
  3. 3.
    Góra, P.: On small stochastic perturbations of the one-sided subshift of the finite type. Bull. Acad. Polon. Sci. 27, No 1, 47–51 (1979)Google Scholar
  4. 4.
    Kaijser, Th.: A Limit Theorem for Markov chains in compact metric spaces with applications to products of random matrices. Duke Math. J. 45, 311–349 (1978)Google Scholar
  5. 5.
    Ruelle, D.: Small Random Perturbations of Dynamical Systems and the Definition of Attractors. Comm. Math. Phys. 82, 137–151 (1981)Google Scholar
  6. 6.
    Shurenkov, V.M.: Ergodic Theorems and Related Topics of Stochastic Processes Theory. Kiev: Naukova Dumka 1981 (in Russian)Google Scholar
  7. 7.
    Smale, S.: The Ω-stability theorem. In: Global Analysis. Proc. Symp. in Pure Math. 14, 289–298. Amer. Math. Soc. Transl. (Providence R.I.) 1970Google Scholar
  8. 8.
    Ventcel, A.D., Freidlin, M.I.: On small stochastic perturbations of dynamical systems. Uspechi Mat. Nauk 25, No 1, 3–55 (1970) (in Russian)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Paweł Góra
    • 1
  1. 1.Institute of MathematicsWarsaw UniversityWarszawaPoland

Personalised recommendations