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Large derivatives, backward contraction and invariant densities for interval maps. (English) Zbl 1138.37019
Summary: We study the dynamics of a smooth multimodal interval map \(f\) with nonflat critical points and all periodic points hyperbolic repelling. Assuming that \(|Df^{n}(f(c))|\rightarrow \infty\) as \(n\rightarrow \infty\) 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 \(\mu\) which is absolutely continuous with respect to Lebesgue measure and the density of \(\mu\) belongs to \(L^{p}\) for all \(p<\ell_{\max}/(\ell_{\max}-1)\), where \(\ell_{\max}\) denotes the maximal critical order of \(f\). In the appendix, we prove that various growth conditions on the derivatives along the critical orbits imply stronger backward contraction.

MSC:
37E05 Dynamical systems involving maps of the interval
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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