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mlc-conjecture
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If true, the Mandelbrot set's whole structure would be described by a simple combinatorial model, pinched disks, and hyperbolicity would be dense among quadratic maps. Yoccoz proved local connectivity at many points, and Jeremy Kahn and Mikhail Lyubich extended it. The full conjecture is open.
Posed 1980s, Adrien Douady and John H. Hubbard.
What would settle it
Lean statement in formal-conjectures