The Lyapunov exponent of holomorphic maps

• 成果类型: Article
• 署名作者: Levin, Genadi; Przytycki, Feliks; Shen, Weixiao
• 署名单位: Hebrew University of Jerusalem; Polish Academy of Sciences; Institute of Mathematics of the Polish Academy of Sciences; National University of Singapore; Fudan University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-015-0637-1
• 发表日期: 2016
• 页码: 363-382
• 关键词: unimodal maps; julia sets; nonuniform hyperbolicity; invariant-measures; rational maps; eckmann; collet; DYNAMICS

摘要

We prove that for any polynomial map with a single critical point its lower Lyapunov exponent at the critical value is negative if and only if the map has an attracting cycle. Similar statement holds for the exponential maps and some other complex dynamical systems. We prove further that for the unicritical polynomials with positive area Julia sets almost every point of the Julia set has zero Lyapunov exponent. Part of this statement generalizes as follows: every point with positive upper Lyapunov exponent in the Julia set of an arbitrary polynomial is not a Lebegue density point.