FROM CANTOR TO SEMI-HYPERBOLIC PARAMETERS ALONG EXTERNAL RAYS

摘要

For the quadratic family f(c)(z) = z(2) + c with c in the exterior of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. Let (c) over cap be a semi-hyperbolic parameter in the boundary of the Mandelbrot set. In this paper we prove that for each z = z(c) in the Julia set, the derivative dz(c)/dc is uniformly O(1/root vertical bar c - (c) over cap vertical bar) when c belongs to a parameter ray that lands on (c) over cap. We also characterize the degeneration of the dynamics along the parameter ray.