From Cantor to semi-hyperbolic parameters along external rays

Abstract

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