Skip to main content
SpringerLink
Log in

Invariant measures for Markov maps of the interval

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

There is a theorem in ergodic theory which gives three conditions sufficient for a piecewise smooth mapping on the interval to admit a finite invariant ergodic measure equivalent to Lebesgue. When the hypotheses fail in certain ways, this work shows that the same conclusion can still be gotten by applying the theorem mentioned to another transformation related to the original one by the method of inducing.

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. Adler, R.L.:F-expansions revisited. Springer lecture notes318, 1–5 (1973)

    Google Scholar 

  2. Adler, R.L.: Continued fractions and Bernoulli Trials. In: Ergodic theory. Moser, J., Phillips, E., Varadhan, S. (eds.). Lecture notes. New York: Courant Inst. Math. Sci. 1975

    Google Scholar 

  3. Adler, R.L., Weiss, B.: The ergodic infinite measure preserving transformation of Boole. Israel J. Math.16, 263–278 (1973)

    Google Scholar 

  4. Bowen, R.: Bernoulli maps of the interval. Israel J. Math.28, 161–168 (1977)

    Google Scholar 

  5. Bunimovich, L.A.: On transformation of the circle. Math. Notes Acad. Sci. USSR8, 204–216 (1970)

    Google Scholar 

  6. Feldman, J.: NewK-automorphisms and a problem of Kakutani. Israel J. Math.24, 16–37 (1976)

    Google Scholar 

  7. Jakobson, M.V., Sinai, Ya.: Oral communication

  8. Lasota, A., Yorke, J.A.: On the existence of invariant measures for piecewise monotonic transformations. Trans. AMS186, 481–488 (1973)

    Google Scholar 

  9. Li, T.Y., Schweiger, F.: The generalized Boole's transformation is ergodic. Manuscripta Math.25, 161–167 (1978)

    Google Scholar 

  10. Ratner, M.: Bernoulli flows over maps of the interval

  11. Renyi, A.: Representations for real numbers and their ergodic properties. Acta Math. Akad. Sci. Hungar.8, 477–493 (1957)

    Google Scholar 

  12. Ruelle, D.: Applications conservant une mesure absolument continue par rapport adx sur [0, 1]. Commun. Math. Phys.55, 47–52 (1977)

    Google Scholar 

  13. Sacksteader, R.: On convergence to invariant measures. Mimeographed notes

  14. Schweiger, F.: Zahlentheoretische Transformationen mit σ-endlichem invarianten Maβ. S.-Ber. Öst. Akad. Wiss. Math.-naturw. Kel. Abt. II.185, 95–103 (1976)

    Google Scholar 

  15. Ulam, S.M., Neumann, J. von: On combination of stochastic and deterministic processes. Bull. AMS53, 1120 (1947)

    Google Scholar 

  16. Wong, S.: Thesis, Berkeley (1977)

  17. Pianigiani, G.: Absolutely continuous invariant measures for the processx n + 1 =Ax n (1 −x n ). Preprint

  18. Sinai, Ya.G.: Introduction to ergodic theory. Princeton: Princeton University Press 1976

    Google Scholar 

  19. Bowen, R., Series, C.: Markov maps associated with Fuchsian groups. I.H.E.S. Publications 50 (1979)

Download references

Authors
  1. Rufus Bowen
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by J.L. Lebowitz

Partially supported by NSF MCS74-19388. A01

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bowen, R. Invariant measures for Markov maps of the interval. Commun.Math. Phys. 69, 1–14 (1979). https://doi.org/10.1007/BF01941319

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01941319

Keywords

Search

Navigation