Speaker: Magnus Aspenberg, associate professor, Department of Mathematics, Lund University, Sweden

Date: May 28, 2024

Time: 14:30-16:00

Location: E119, Huagangyuan Building, Qingdao Campus

Sponsor: Frontiers Science Center for Nonlinear Expectations, Shandong University; Research Centre for Mathematics and Interdisciplinary Sciences Centre, Shandong University; Sino-Russian Mathematics Center in Qingdao

Abstract:

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.

For more information, please visit:

https://www.view.sdu.edu.cn/info/1020/191578.htm