Parabolic Towers and the Asymptotic Geometry of the Mandelbrot Set

Abstract

Understanding the geometry of the Mandelbrot set has been a central pillar of holomorphic dynamics over the past four decades. Much of its structure is now understood, but a critical question remains unresolved: is the Mandelbrot set locally connected? The first major break- through towards this conjecture was achieved by Yoccoz in the nineties, who proved that the Mandelbrot set is locally connected at all parameters which are not infinitely quadratic-like renormalizable. A key ingredient in Yoccoz’s work is the PLY-inequality, which bounds the diameter of certain subsets, called limbs, of the Mandelbrot set. These limbs are naturally labeled by the rational numbers, and the PLY-inequality asserts that the p/q-limb of the Mandelbrot set has size O(1/q). Milnor conjectured that O(1/q2) is the correct scale. For any N ≥ 1, the main result of this thesis is to verify Milnor’s conjecture for all p/q-limbs where a finite continued fraction of p/q has uniformly bounded length. Our strategy relies on careful analysis of the bifurcation of parabolic fixed points; we also further develop some of the classical theory in this area. We introduce parabolic and near-parabolic renormalization operators for maps which have parabolic fixed points of arbitrary multiplier and there perturbations, constructing invariant classes for these operators. We provide an alternative definition to the parabolic towers introduced by Epstein and construct a dynamically natural topology on the space of all parabolic towers. We also study the dynamics of Lavaurs maps, constructing analogues of polynomial external rays for these functions showing that these rays arise as the Hausdorff limits of polynomial external rays.