From hyperbolic to parabolic parameters along internal rays

Abstract

For the quadratic family f c ( z ) = z 2 + c f_{c}(z) = z^2+c with c c in a hyperbolic component of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. In this paper we give a uniform derivative estimate of such a motion when the parameter c c converges to a parabolic parameter c ^ {\hat {c}} radially; in other words, it stays within a bounded Poincaré distance from the internal ray that lands on c ^ {\hat {c}} . We also show that the motion of each point in the Julia set is uniformly one-sided Hölder continuous at c ^ {\hat {c}} with exponent depending only on the petal number. This paper is a parabolic counterpart of the authors’ paper “From Cantor to semi-hyperbolic parameters along external rays” [Trans. Amer. Math. Soc. 372 (2019), pp. 7959–7992].