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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.

37E05 Dynamical systems involving maps of the interval
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
Full Text: DOI
[1] Bruin, H.: The existence of absolutely continuous invariant measures is not a topological invariant for unimodal maps. Ergodic Theory Dyn. Syst. 18(3), 555–565 (1998) · Zbl 0928.37004
[2] Bruin, H.: Invariant measures of interval maps. Ph.D. Thesis, University of Delft (1994) · Zbl 0812.58052
[3] Bruin, H., Keller, G., Nowicki, T., van Strien, S.: Wild attractors exist. Ann. Math. 143(1), 97–130 (1996) · Zbl 0848.58016
[4] Bruin, H., Shen, W., van Strien, S.: Invariant measure exists without a growth condition. Commun. Math. Phys. 241, 287–306 (2003) · Zbl 1098.37034
[5] Bruin, H., van Strien, S.: Existence of invariant measures for multimodal interval maps. In: Global Analysis of Dynamical Systems, pp. 433–447. Inst. Phys., Bristol (2001) · Zbl 1198.37066
[6] Collet, P., Eckmann, J.-P.: Positive Liapunov exponents and absolutely continuity for maps of the interval. Ergodic Theory Dyn. Syst. 3(1), 13–46 (1983) · Zbl 0532.28014
[7] Graczyk, J., Sands, D.: Manuscript in preparation
[8] Misiurewicz, M.: Absolutely continuous measures for certain maps of an interval. Publ. Math., Inst. Hautes ’Etud. Sci. 53, 17–51 (1981) · Zbl 0477.58020
[9] de Melo, W., van Strien, S.: One-Dimensional Dynamics. Springer, Berlin (1993) · Zbl 0791.58003
[10] Nowicki, T., van Strien, S.: Hyperbolicity properties of C 2 multi-modal Collet–Eckmann maps without Schwarzian derivative assumptions. Trans. Am. Math. Soc. 321(2), 793–810 (1990) · Zbl 0731.58021
[11] Nowicki, T., van Strien, S.: Invariant measures exist under a summability condition for unimodal maps. Invent. Math. 105, 123–136 (1991) · Zbl 0736.58030
[12] Rivera-Letelier, J.: A connecting lemma for rational maps satisfying a no growth condition. Ergodic Theory Dyn. Syst. 27(2), 595–636 (2007) · Zbl 1110.37037
[13] van Strien, S., Vargas, E.: Real Bounds, ergodicity and negative Schwarzian for multimodal maps. J. Am. Math. Soc. 17(4), 749–782 (2004) · Zbl 1073.37043
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