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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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References:
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