Communications in Mathematical Physics

, Volume 81, Issue 3, pp 437–453

Analyticity properties of the Feigenbaum function

  • H. Epstein
  • J. Lascoux


Analyticity properties of the Feigenbaum function [a solution ofg(x)=−λ−1g(gx)) withg(0)=1,g′(0)=0,g″(0)<0] are investigated by studying its inverse function which turns out to be Herglotz or anti-Herglotz on all its sheets. It is found thatg is analytic and uniform in a domain with a natural boundary.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Campanino, M., Epstein, H., Ruelle, D.: Topology (to appear); Campanino, M., Epstein, H.: Commun. Math. Phys.79, 261–302 (1981)Google Scholar
  2. 2.
    Collet, P., Eckmann, J.-P.: Iterated maps on the interval as dynamical systems. Boston: Birkhäuser 1980Google Scholar
  3. 3.
    Collet, P., Eckmann, J.-P., Lanford III, O.E.: Commun. Math. Phys.76, 211–254 (1980)Google Scholar
  4. 4.
    Coullet, P., Tresser, C.: C.R.A.S. Paris287A, 577 (1978); J. Phys. Coll.39, C5–25 (1978)Google Scholar
  5. 5.
    Feigenbaum, M.J.: J. Stat. Phys.19, 25–52 (1978);21, 669–706 (1979)Google Scholar
  6. 6.
    Feigenbaum, M.J.: Commun. Math. Phys.77, 65–86 (1980)Google Scholar
  7. 7.
    Lanford III, O.E.: Remarks on the accumulation of period-doubling bifurcations. In: Mathematical problems in theoretical physics. Proceedings, Lausanne 1979. Berlin, Heidelberg, New York: Springer 1980Google Scholar
  8. 8.
    Lanford III, O.E.: Smooth transformations of intervals. Séminaire Bourbaki 1980/81, No. 563Google Scholar
  9. 9.
    Lanford III, O.E.: A computer-assisted proof of the Feigenbaum conjectures. IHES Preprint P/81/17 (to appear). Also: Private communication, and article in preparationGoogle Scholar
  10. 10.
    Lanford III, O.E.: Seminar at I.H.E.S., Bures-sur-Yvette, 1981 (to appear)Google Scholar
  11. 11.
    Rudin, W.: Real and complex analysis. New York: McGraw-Hill 1966Google Scholar
  12. 12.
    Valiron, G.: Fonctions analytiques. Paris: Presses Universitaires de France 1954Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • H. Epstein
    • 1
  • J. Lascoux
    • 2
  1. 1.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance
  2. 2.Centre de Physique Théorique, Ecole PolytechniquePalaiseauFrance

Personalised recommendations