Large derivatives, backward contraction and invariant densities for interval maps

• 成果类型: Article
• 署名作者: Bruin, H.; Rivera-Letelier, J.; Shen, W.; van Strien, S.
• 署名单位: University of Surrey; Universidad Catolica del Norte; Chinese Academy of Sciences; University of Science & Technology of China, CAS; University of Warwick
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-007-0108-4
• 发表日期: 2008
• 页码: 509-533
• 关键词: growth condition; schwarzian

摘要

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 vertical bar Df(n)(f(c))vertical bar -> infinity as n -> infinity 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 < l(max)/(l(max) - 1), where l(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.