From Hyperbolic to Parabolic Parameters along Internal Rays

For the quadratic family $f_c(z)=z^2+c$ with $c$ 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 $c$ converges to a parabolic parameter $\widehat{c}$ radially; in other words, it stays within a bounded Poincaré distance from the internal ray that lands on $\widehat{c}$. We also show that the motion of each point in the Julia set is uniformly one-sided Hölder continuous at $\widehat{c}$ 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).