Magnus Aspenberg: Collet-Eckmann maps in the unicritical family

Abstract: The Collet-Eckmann condition is one, quite strong, condition which is used to exhibit chaotic behaviour. 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 f_c(z) = z^d + c, where c is a parameter in the complex plane. This is a well studied family of maps and it has a long history. Let Md be the connectedness locus, or simply, the Mandelbrot set, i.e. parameters c for which the Julia set is connected. Around the millenia shift, J. Rivera-Letelier proved that critically non-recurrent maps (Misiurewicz maps) in this family are Lebesgue density points of the complement of Md (I proved a corresponding result for rational maps in 2009). In a series of quite recent papers, J. Graczyk and G. Swiatek proved, among other things, that typical CE-parameters w.r.t. harmonic measure are Lebesgue density points of the complement of Md. 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 fc in the unicritical family, c 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.