Perturbations of Collet-Eckmann maps in the unicritical family

一、报告题目 Perturbations of Collet-Eckmann maps in the unicritical family 二、主讲人 Magnus Aspenberg（Lund University） 三、报告时间 2024年5月28日 14:30–16:00 四、报告地点 青岛校区华岗苑东楼E119 五、摘要 The Collet-Eckmann condition is one, quite strong, condition which is used to exhibit chaotic behaviour. For instance, it implies the existence of an absolutely continuous invariant measure in many families of maps. In this talk I will present a recent result (joint work with M. Bylund and W. Cui) about perturbations of CE-maps in the unicritical family. This is a well studied family of maps and it has a long history. Let M_d be the connectedness locus, or simply, the Mandelbrot set, i.e. parameters for which the Julia set is connected. Around the millenia shift, J. Rivera-Letelier proved that critically non-recurrent maps in this family are Lebesgue density point of the complement of M_d (I proved a corresponding result for rational maps in 2009). In a series of quite recent papers, J. Graczyk and G. Swiatek proves, among other things, that typical CE-parameters w.r.t. harmonic measure are Lebesgue density points of the complement of M_d. For these maps, in particular, the critical point is allowed to be slowly recurrent. Moreover, in 2011, A. Avila, M. Lyubich and W. Shen proved that, in particular, CE-maps cannot be density points of Md. The main result I will present is that for each CE-map in the unicritical family is a Lebesgue density point of the complement of the Mandelbrot set. It also generalizes earlier results by M. Bylund W. Cui and myself for slowly recurrent rational maps. 六、主讲人简介 Magnus Aspenberg，瑞典隆德大学数学系副教授。主要从事复动力系统的研究，在PLMS, CMP, Math. Ann.，TAMS等期刊上发表多篇论文。 七、主办单位 非线性期望前沿科学中心 数学与交叉科学研究中心 中俄数学中心青岛基地