Communications in Mathematical Physics

, Volume 241, Issue 2–3, pp 287–306 | Cite as

Invariant Measures Exist Without a Growth Condition



Given a non-flat S-unimodal interval map f, we show that there exists C which only depends on the order of the critical point c such that if |Dfn(f(c))|≥C for all n sufficiently large, then f admits an absolutely continuous invariant probability measure (acip). As part of the proof we show that if the quotients of successive intervals of the principal nest of f are sufficiently small, then f admits an acip. As a special case, any S-unimodal map with critical order ℓ<2+ɛ having no central returns possesses an acip. These results imply that the summability assumptions in the theorems of Nowicki & van Strien [21] and Martens & Nowicki [17] can be weakened considerably.


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  1. 1.
    Bowen, R.: Invariant measures for Markov maps of the interval. Commun. Math. Phys. 69, 1–17 (1979)MathSciNetMATHGoogle Scholar
  2. 2.
    Bruin, H.: Topological conditions for the existence of invariant measures for unimodal maps. Ergod. Th. & Dynam. Sys. 14, 433–451 (1994)Google Scholar
  3. 3.
    Bruin, H.: Topological conditions for the existence of Cantor attractors. Trans. Am. Math. Soc. 350, 2229–2263 (1998)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Bruin, H., Keller, G., Nowicki, T., van Strien, S.: Wild Cantor attractors exist. Ann. Math. 143, 97–130 (1996)MathSciNetMATHGoogle Scholar
  5. 5.
    Collet, P., Eckmann, J.-P.: Positive Liapunov exponents and absolute continuity for maps of the interval. Ergod. Th. & Dyn. Sys. 3, 13–46 (1983)Google Scholar
  6. 6.
    Graczyk, J., Świtek, G.: Induced expansion for quadratic polynomials. Ann. Sci Éc. Norm. Súp. 29, 399–482 (1996)MATHGoogle Scholar
  7. 7.
    Graczyk, J., Sands, D., Świtek, G.: Decay of geometry for unimodal maps: Negative Schwarzian case. Preprint, 2000Google Scholar
  8. 8.
    Jakobson, M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81, 39–88 (1981)MathSciNetMATHGoogle Scholar
  9. 9.
    Jakobson, M., Świtek, G.: Metric properties of non-renormalizable S-unimodal maps. I. Induced expansion and invariant measures. Ergod. Th. & Dynam. Sys. 14, 721–755 (1994)Google Scholar
  10. 10.
    Johnson, S.: Singular measures without restrictive intervals. Commun. Math. Phys. 110, 185–190 (1987)MathSciNetMATHGoogle Scholar
  11. 11.
    Keller, G., Nowicki, T.: Fibonacci maps re(aℓ)visited. Ergod. Th. & Dyn. Sys. 15, 99–120 (1995)Google Scholar
  12. 12.
    Lyubich, M.: Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. of Math. 140, 347–404 (1994) and Erratum Manuscript, 2000MathSciNetMATHGoogle Scholar
  13. 13.
    Lyubich, M., Milnor, J.: The Fibonacci unimodal map. J. Am. Math. Soc. 6, 425–457 (1993)MathSciNetMATHGoogle Scholar
  14. 14.
    Mañé, R.: Hyperbolicity, sinks and measure in one-dimensional dynamics. Commun. Math. Phys. 100(4), 495–524 (1985)Google Scholar
  15. 15.
    Martens, M.: Interval dynamics. Ph.D. Thesis, Delft, 1990Google Scholar
  16. 16.
    Martens, M.: Distortion results and invariant Cantor sets of unimodal maps. Ergod. Th. & Dynam. Sys. 14, 331–349 (1994)Google Scholar
  17. 17.
    Martens, M., Nowicki, T.: Invariant measures for typical quadratic maps, Géométrie complexe et systèmes dynamiques (Orsay, 1995). Astérisque 261, 239–252 (2000)MATHGoogle Scholar
  18. 18.
    de Melo, W., van Strien, S.: One-dimensional dynamics. Berlin-Heidelberg-New York: Springer, 1993Google Scholar
  19. 19.
    Misiurewicz, M.: Absolutely continuous measures for certain maps of an interval. Publ. Math. I.H.E.S. 53, 17–51 (1981)MathSciNetMATHGoogle Scholar
  20. 20.
    Nowicki, T.: A positive Liapunov exponent for the critical value of an S-unimodal mapping implies uniform hyperbolicity. Ergod. Th. & Dynam. Sys. 8, 425–435 (1988)Google Scholar
  21. 21.
    Nowicki, T., van Strien, S.: Invariant measures exist under a summability condition. Invent. Math. 105, 123–136 (1991)MathSciNetMATHGoogle Scholar
  22. 22.
    Pianigiani, G.: Absolutely continuous invariant measures on the interval for the process x n+1=Ax n(1-x n). Boll. Un. Mat. Ital. 16, 364–378 (1979)Google Scholar
  23. 23.
    Shen, W.: Decay geometry for unimodal maps: An elementary proof. Preprint Warwick, 2002Google Scholar
  24. 24.
    Straube, E.: On the existence of invariant absolutely continuous measures. Commun. Math. Phys. 81, 27–30 (1981)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Henk Bruin
    • 1
  • Weixiao Shen
    • 2
  • Sebastian van Strien
    • 2
  1. 1.Department of MathematicsUniversity of GroningenAV GroningenThe Netherlands
  2. 2.Department of MathematicsUniversity of WarwickCoventry CV4 7ALUK

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