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arXiv · 2202.09832

Surgery sequences and self-similarity of the Mandelbrot set

Danny Calegari
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Abstract

We introduce an analog in the context of rational maps of the idea of hyperbolic Dehn surgery from the theory of Kleinian groups. A surgery sequence is a sequence of postcritically finite maps limiting (in a precise manner) to a postcritically finite map with at least one strictly preperiodic critical orbit. As an application of this idea we give a new and elementary proof of Tan Lei's theorem on the asymptotic self-similarity of Julia Sets and the Mandelbrot Set at Misiurewicz points.

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Danny Calegari. 2024-08-27. Surgery sequences and self-similarity of the Mandelbrot set. https://doi.org/10.2140/agt.2025.25.2807

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