Collet, Eckmann and the bifurcation measure

• 成果类型: Article
• 署名作者: Astorg, Matthieu; Gauthier, Thomas; Mihalache, Nicolae; Vigny, Gabriel
• 署名单位: Universite de Orleans; Universite de Picardie Jules Verne (UPJV); Universite Paris Saclay; Institut Polytechnique de Paris; Ecole Polytechnique; Centre National de la Recherche Scientifique (CNRS); Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00874-5
• 发表日期: 2019
• 页码: 749-797
• 关键词: mandelbrot set; rational maps; DYNAMICS; currents; Iterations; families

摘要

The moduli space M-d of degree d >= 2 rational maps can naturally be endowed with a measure mu(bif) detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure mu bif has positive Lebesgue measure. To do so, we establish a general sufficient condition for the conjugacy class of a rational map to belong to the support of mu bif and we exhibit a large set of Collet-Eckmann rational maps which satisfy this condition. As a consequence, we get a set of Collet-Eckmann rational maps of positive Lebesgue measure which are approximated by hyperbolic rational maps.