The Collet-Eckmann condition plays an important role in the study of both real and rational dynamics. This condition, which requires exponential increase of the derivative along the critical orbit(s), is known to be abundant in the real quadratic setting, and also in the rational setting. For complex quadratic maps, the set of Collet-Eckmann parameter are known to constitute a set of zero Lebesgue measure (area), but full harmonic measure.
An important open question in complex dynamics is the Fatou conjecture: Does the set of hyperbolic complex quadratic maps constitute a dense set? In this talk I will discuss this question in a strong form around Collet-Eckmann parameters: Every unicritical Collet-Eckmann parameter is a Lebesgue density point of the complement of the Mandelbrot set. This talk is based on joint work with Magnus Aspenberg and Weiwei Cui.
