Multipliers of periodic orbits of quadratic polynomials and the parameter plane

We prove a result about an extension of the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove that the size of p/q-limb of a hyperbolic component of the Mandelbrot set of period n is O(4 ⁿ /p), and give an explicit condition on internal arguments under which the Julia set of corresponding (unique) infinitely renormalizable quadratic polynomial is not locally connected.