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Holomorphic Removability of Julia Sets

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arxiv math/9812164 v1 pith:Q3GJHRKT submitted 1998-12-31 math.DS math.CV

Jeremy Kahn This is my paper · ORCID
classification math.DSmath.CV
keywords complexconformaljuliamandelbrotplaneassumeconnectedcorollary
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abstract

Let $f(z) = z^2 + c$ be a quadratic polynomial, with c in the Mandelbrot set. Assume further that both fixed points of f are repelling, and that f is not renormalizable. Then we prove that the Julia set J of f is holomorphically removable in the sense that every homeomorphism of the complex plane to itself that is conformal off of J is in fact conformal on the entire complex plane. As a corollary, we deduce that the Mandelbrot Set is locally connected at such c.

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Cited by 1 Pith paper

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  1. MLC for parabolically bounded primitive renormalization

    math.DS 2026-06 unverdicted novelty 6.0 of 10

    Proves a priori bounds and MLC for parabolically bounded primitive renormalization of the Mandelbrot set via new tools controlling renormalization degeneration.

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