Communications in Mathematical Physics

, Volume 241, Issue 2, pp 287-306

First online:

Invariant Measures Exist Without a Growth Condition

  • Henk BruinAffiliated withDepartment of Mathematics, University of Groningen Email author 
  • , Weixiao ShenAffiliated withDepartment of Mathematics, University of Warwick
  • , Sebastian van StrienAffiliated withDepartment of Mathematics, University of Warwick

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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.