Large derivatives,backward contraction and invariant densities for interval maps |
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Authors: | Henk Bruin Juan Rivera-Letelier Weixiao Shen Sebastian van Strien |
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Institution: | (1) Department of Mathematics, Faculty of Engeneering and Physical Sciences, University of Surrey, Guildford, GU2 7XH, UK;(2) Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile;(3) Department of Mathematics, University of Sience and Technology, Hefei, 230026, P.R. China;(4) Department of Mathematics, University of Warwick, Coventry, CV4 7AL, UK |
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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<ℓmax/(ℓmax-1), where ℓmax 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. |
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