Institute for Mathematical Sciences
Holomorphic Removability of Julia Sets
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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