Skip to main content
SpringerLink
Log in

Concerning a stochastic dynamical system

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We study the discrete-time dynamical system

$$X_{n + 1} = 2\sigma \cos (2\pi \theta _n )g(X_n ), n \in \mathbb{Z},$$

Whereθ n is an ergodic stationary process whose univariate distribution is uniform on the interval [0, 1], the functiong(x) is odd, bounded, increasing, and continous, and ℤ is the ring of integers. It is proved that under certain conditions there exists a unique stationary process that is a solution of the above equation and this process has a continous purely singular spectrum.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. S. Pikovski and U. Feudel, “Correlations and spectra of strange nonchaotic attractors”,J. Phys. A.,27, 5209–5219 (1994).

    Google Scholar 

  2. A. S. Pikovski, U. Feudel, J. Kurths, and M. A. Zaks, “Singular continuous spectra in dissipative dynamics,” in:Nichtlineare Dynamik, Preprint, Max-Planck-Arbeitsgruppe (1994), pp. 1–25.

  3. V. I. Oseledets, “The spectra of ergodic automorphisms”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.]168, No. 5, 1009–1011 (1968).

    Google Scholar 

  4. I. P. Kornfel'd, Ya. G. Sinai, and S. P. Fomin,Ergodic Theory [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  5. A. V. Katok and A. M. Stepin, “Approximation in ergodic theory”Uspekhi Math. Nauk [Russian Math. Surveys],49, No. 2, 143–144 (1994).

    Google Scholar 

  6. O. N. Ageev, “Mixing in the components of the rearrangementT α,β ”Uspekhi Mat. Nauk [Russian Math. Surveys]49, No. 2, 143–144 (1994).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State Institute of Electronics and Mathematics, Moscow, USSR

    Z. I. Bezhaeva

  2. M. V. Lomonosov Moscow State University, Moscow, USSR

    V. I. Oseledets

Authors
  1. Z. I. Bezhaeva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. I. Oseledets
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 803–809, June, 1997.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bezhaeva, Z.I., Oseledets, V.I. Concerning a stochastic dynamical system. Math Notes 61, 675–680 (1997). https://doi.org/10.1007/BF02361208

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02361208

Key words

Search

Navigation