Surgery sequences and self-similarity of the Mandelbrot set

Calegari, Danny

doi:10.2140/agt.2025.25.2807

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ISSN (electronic): 1472-2739

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Surgery sequences and self-similarity of the Mandelbrot set

Danny Calegari

Algebraic & Geometric Topology 25 (2025) 2807–2816

DOI: 10.2140/agt.2025.25.2807

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.

For Tan Lei

Keywords

Mandelbrot set, Julia set, Misiurewicz point, Tan Lei's theorem, self-similarity, surgery sequence, Sullivan's dictionary

Mathematical Subject Classification

Primary: 37F10

References

Publication

Received: 25 February 2022

Revised: 30 July 2024

Accepted: 22 August 2024

Published: 17 September 2025

Authors

Danny Calegari
Department of Mathematics University of Chicago Chicago, IL United States

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Volume 25, issue 5 (2025)