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Invariant measures exist without a growth condition. (English) Zbl 1098.37034
Summary: 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 \(| Df^n(f(c))|\geq C\) 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 \(\ell<2+\varepsilon\) having no central returns possesses an acip. These results imply that the summability assumptions in the theorems of T. Nowicki and S. van Strien [Invent. Math. 105, No. 1, 123–136 (1991; Zbl 0736.58030)] and M. Martens and T. Nowicki [Astérisque 261, 239–252 (2000; Zbl 0939.37020)] can be weakened considerably.

MSC:
37E05 Dynamical systems involving maps of the interval
37A05 Dynamical aspects of measure-preserving transformations
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