Skip to main content
SpringerLink
Log in

Universal properties of maps of the circle with ɛ-singularities

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

Abstract

Following the work of Collet, Eckmann, and Lanford on the Feigenbaum conjecture, we study the structure of the renormalization transformation introduced in [12] upon maps of the circle with critical points of the formx|x|ɛ.

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. Arnold, V.I.: Small denominators I; on the mapping of the circumference onto itself. Izv. Akad. Nauk.25, 21–86 (1961) [Transl. Am. Math. Soc., 2nd Ser.46, 213–284 (1965)]

    Google Scholar 

  2. Brunovsky, P.: Generic properties of the rotation number of one-parameter diffeomorphisms of the circle. Czech. Math. J.24, 74–90 (1974)

    Google Scholar 

  3. Collet, P., Eckmann, J.-P., Lanford III, O.E.: Universal properties of maps on an interval. Commun. Math. Phys.76, 211–254 (1980)

    Google Scholar 

  4. Feigenbaum, M.J., Kadanoff, L.P., Shenker, S.J.: Quasiperiodicity in dissipative systems: a renormalization group analysis. Physica D (1982) (submitted)

  5. Herman, M.R.: Mesure de Lebesque et nombre de rotation, geometry and topology. In: Lecture Notes in Mathematics, Vol. 597. Berlin, Heidelberg, New York: Springer 1977, pp. 271–293

    Google Scholar 

  6. Herman, M.R.: Sur la conjugaison différentiable des difféomorphismes du cercle a des rotations. I.H.E.S. Publ. Math.49, 5–234 (1979)

    Google Scholar 

  7. Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant manifolds. In: Lecture Notes in Mathematics, Vol. 583. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  8. Kadanoff, L.P.: An ɛ-expansion for a one-dimensional map. Preprint (1982)

  9. Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966

    Google Scholar 

  10. Lanford III, O.E.: A computer-assisted proof of the Feigenbaum conjectures. Bull. Am. Math. Soc.6, 427–434 (1982)

    Google Scholar 

  11. Lang, S.: Differential manifolds. Reading, MA: Addison-Wesley 1972

    Google Scholar 

  12. Rand, D., Ostlund, S., Sethna, J., Siggia, E.D.: A universal transition from quasi-periodicity to chaos in dissipative systems. Phys. Rev. Lett.49, 132–135 (1982), and Universal Properties of bifurcations from quasiperiodicity in dissipative systems. ITP Preprint 1982

    Google Scholar 

  13. Shenker, S.J.: Scaling behaviour in a map of a circle onto itself: empirical results. Physica D (1982) (to be published)

Download references

Author information

Author notes

  1. Leo Jonker

    Present address: Department of Mathematics and Statistics, Queen's University, K7L 3N6, Kingston, Ontario, Canada

  2. David A. Rand

    Present address: Mathematics Institute, University of Warwick, CV4 7AL, Coventry, England

Authors and Affiliations

  1. Institute for Theoretical Physics, University of California, 93106, Santa Barbara, CA, USA

    Leo Jonker & David A. Rand

Authors
  1. Leo Jonker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David A. Rand
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by O. E. Lanford

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jonker, L., Rand, D.A. Universal properties of maps of the circle with ɛ-singularities. Commun.Math. Phys. 90, 273–292 (1983). https://doi.org/10.1007/BF01205508

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01205508

Keywords

Search

Navigation