Inventiones mathematicae

, Volume 105, Issue 1, pp 123–136

Invariant measures exist under a summability condition for unimodal maps

  • Tomasz Nowicki
  • Sebastian van Strien
Article

Summary

For unimodal maps with negative Schwarzian derivative a sufficient condition for the existence of an invariant measure, absolutely continuous with respect to Lebesgue measure, is given. Namely the derivatives of the iterations of the map in the (unique) critical value must be so large that the sum of (some root of) the inverses is finite.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B-Y] Benedicks, M., Young, L.S.: Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps. Preprint (1990)Google Scholar
  2. [BL1] Blokh, A.M., Lyubich, M.Yu.: Attractors of maps of the interval. Funct. Anal. Appl.21, 70–71 (1987) (in Russian)Google Scholar
  3. [BL2] Blokh, A.M., Lyubich, M.Yu.: Measurable dynamics of S-unimodal maps of the interval. Preprint (1989)Google Scholar
  4. [C-E] Collet, P., Eckmann, J.-P.: Positive Liapounov exponents and absolute continuity for maps of the interval. Ergodic Theory Dyn. Syst.3, 13–46 (1983)Google Scholar
  5. [Ja] Jakobson, M.: Absolutely continuous measures for one-parameter families of one-dimensional maps. Comm. Math. Phys.81, 39–88 (1981)Google Scholar
  6. [Jo] Johnson, S.: Singular measures without restrictive intervals. Comm. Math. Phys.110, 655–659 (1987)Google Scholar
  7. [K1] Keller, G.: Invariant measures and Lyapounov exponents for S-unimodal maps. Preprint (1987)Google Scholar
  8. [K2] Keller, G.: Exponents, attractors, and Hopf decompositions for interval maps. Preprint (1988)Google Scholar
  9. [L.Y] Lasota, A., Yorke, J.: On the existence of invariant measures for piecewise monotonic transformations. Trans. A.M.S.186, 481–488 (1973)Google Scholar
  10. [M.S.1] de Melo, W., van Strien, S.: A structure theorem in one-dimensional dynamics, Ann. Math.129, 519–546 (1989)Google Scholar
  11. [M.S.2] de Melo, W., van Strien, S.: One-dimensional dynamics, to appearGoogle Scholar
  12. [M.M.S.] Martens, M., de Melo, W., van Strien, S.: Julia-Fatou-Sullivan theory for real one-dimensional dynamics. Acta Math. (To appear)Google Scholar
  13. [M] Misiurewicz, M.: Absolutely continuous measure for certain maps of an interval. Publ. I.H.E.S.53, 17–51 (1981)Google Scholar
  14. [N1] Nowicki, T.: Symmetric S-unimodal properties and positive Liapounov exponents, Ergodic Theory Dyn. Syst.5, 611–616 (1985)Google Scholar
  15. [N2] Nowicki, T.: A positive Liapounov exponent of the critical value of S-unimodal maps implies uniform hyperbolicity. Ergodic Theory Dyn. Syst.8, 425–435 (1988)Google Scholar
  16. [N-S] Nowicki, T., van Strien, S.J.: Absolutely continuous invariant measures for C2 maps satisfying the Collet-Eckmann conditions. Invent. Math.93, 619–635 (1988)Google Scholar
  17. [Str1] van Strien, S.J.: On the creation of horseshoes. Lect. Notes Math.898, 316–351 (1981)Google Scholar
  18. [Str2] van Strien, S.J.: Hyperbolicity and invariant measures for general C2 interval maps satisfying the Misiurewicz condition. Commun. Math Phys.128, 437–496 (1990)Google Scholar
  19. [Sz] Szlenk, W.: Some dynamical properties of certain differentiable mappings. Bol. Soc. Mat. Mex., II Ser.24, 57–87 (1979)Google Scholar

Copyright information

© Springer International 1991

Authors and Affiliations

  • Tomasz Nowicki
    • 1
  • Sebastian van Strien
    • 2
  1. 1.Mathematics DepartmentWarsaw UniversityPoland
  2. 2.Mathematics DepartmentTU DelftThe Netherlands

Personalised recommendations