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


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.


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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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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