國立清華大學 數學系

Abstract For the quadratic family f_c (z) = z^2+c with c in the exterior of the Mandelbrot set, it is well-known that every point in the Julia set moves holomorphically when c varies. Let \hat{c} be a semi-hyperbolic parameter in the boundary of the Mandelbrot set. In this talk, I shall show that for each z = z(c) in the Julia set, the derivative dz(c)/dc is uniformly O(1/\sqrt{|c-\hat{c}|}) when c belongs to a parameter ray that lands on \hat{c}. I shall also discuss the degeneration of the dynamics along the parameter ray.