Invariant Measures Exist Without a Growth Condition
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- Bruin, H., Shen, W. & Strien, S. Commun. Math. Phys. (2003) 241: 287. doi:10.1007/s00220-003-0928-z
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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 |Dfn(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  and Martens & Nowicki  can be weakened considerably.