From Cantor to Semi-hyperbolic Parameter along External Rays

TL;DR

This paper investigates the behavior of Julia sets and their derivatives along external rays approaching semi-hyperbolic parameters on the Mandelbrot set boundary, revealing uniform bounds and dynamic degeneration patterns.

Contribution

It establishes a uniform bound on the derivative of Julia set points with respect to parameters near semi-hyperbolic points and characterizes the dynamic degeneration along parameter rays.

Findings

Derivative of Julia points is O(1/√|c−ĉ|) near semi-hyperbolic parameters.

Dynamics degenerate in a specific manner along parameter rays landing on semi-hyperbolic points.

Results apply to parameters outside the Mandelbrot set with holomorphic Julia set movement.

Abstract

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 $\hat{c}$ be a semi-hyperbolic parameter in the boundary of the Mandelbrot set. In this paper we prove that for each $z = z(c)$ 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 $\hat{c}$. We also characterize the degeneration of the dynamics along the parameter ray.