  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.
