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Communications in Mathematical Physics

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

Invariant Measures Exist Without a Growth Condition

  • Henk BruinEmail author
  • Weixiao Shen
  • Sebastian van Strien
Article

Abstract

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 |Df n (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.

Keywords

Growth Condition Probability Measure Invariant Measure Central Return Invariant Probability Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bowen, R.: Invariant measures for Markov maps of the interval. Commun. Math. Phys. 69, 1–17 (1979)MathSciNetzbMATHGoogle 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)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Bruin, H., Keller, G., Nowicki, T., van Strien, S.: Wild Cantor attractors exist. Ann. Math. 143, 97–130 (1996)MathSciNetzbMATHGoogle 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)zbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle 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, 2000MathSciNetzbMATHGoogle Scholar
  13. 13.
    Lyubich, M., Milnor, J.: The Fibonacci unimodal map. J. Am. Math. Soc. 6, 425–457 (1993)MathSciNetzbMATHGoogle 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)zbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Henk Bruin
    • 1
    Email author
  • 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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