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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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##### References:
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