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.

48 pages. 9 PostScript figures

References in corpus (4)

• Dynamics in one complex variable: introductory lectures
• Local connectivity of Julia sets: expository lectures
• On the Lebesgue measure of the Julia set of a quadratic polynomial
• Geometry of quadratic polynomials: moduli, rigidity and local connectivity

Cited by in corpus (8)

• Statistical properties of one-dimensional maps under weak hyperbolicity assumptions
• Non-removability of the Sierpinski Gasket
• Geometrically finite transcendental entire functions
• Combinatorial rigidity for unicritical polynomials
• Primitive tuning via quasiconformal surgery
• Primitive tuning for non-hyperbolic polynomials
• Removability and non-injectivity of conformal welding
• Parapuzzle of the Multibrot set and typical dynamics of unimodal maps