The near-parabolic geometry of external rays

Using Lavaurs maps and near-parabolic renormalization, we describe the degenerating geometry of external rays for quadratic polynomials when a periodic cycle becomes parabolic. We similarly describe the geometry of parameter rays for the Mandelbrot set near parabolic points. Using this geometric control we establish new bounds on the size of limbs of the Mandelbrot set, for example providing a quadratic Pommerenke-Levin-Yoccoz inequality in the near-parabolic setting.