Large derivatives, backward contraction and invariant densities for interval maps

Abstract

In this paper, we study the dynamics of a smooth multimodal interval map f with non-flat critical points and all periodic points hyperbolic repelling. Assuming that |Dfn(f(c))|→∞ as n→∞ holds for all critical points c, we show that f satisfies the so-called backward contracting property with an arbitrarily large constant, and that f has an invariant probability μ which is absolutely continuous with respect to Lebesgue measure and the density of μ belongs to Lp for all p