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arXiv · 1803.03130

From Cantor to Semi-hyperbolic Parameter along External Rays

Yi-Chiuan Chen·Tomoki Kawahira
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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.

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Yi-Chiuan Chen, Tomoki Kawahira. 2018-03-08. From Cantor to Semi-hyperbolic Parameter along External Rays. https://arxiv.org/abs/1803.03130

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