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Pacman renormalization and self-similarity of the Mandelbrot set near Siegel parameters


Authors: Dzmitry Dudko, Mikhail Lyubich and Nikita Selinger
Journal: J. Amer. Math. Soc. 33 (2020), 653-733
MSC (2010): Primary 37E20, 37F25, 37F45
DOI: https://doi.org/10.1090/jams/942
Published electronically: June 16, 2020
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Abstract: In the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of ``Pacman Renormalization Theory'' that combines features of quadratic-like and Siegel renormalizations. We show that Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to pacman renormalization periodic points. Then we prove that these periodic points are hyperbolic with one-dimensional unstable manifold, resolving a long-standing problem. As a consequence, we obtain the scaling laws for the centers of satellite components of the Mandelbrot set near the corresponding Siegel parameters.


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Additional Information

Dzmitry Dudko
Affiliation: Mathematisches Institut, Universitat Gottingen, 37073 Gottingen, Germany
Address at time of publication: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794
Email: dzmitry.dudko@stonybrook.edu

Mikhail Lyubich
Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794
Email: mlyubich@math.stonybrook.edu

Nikita Selinger
Affiliation: Department of Mathematics, University of Alabama at Birmingham, 4005 University Hall, 1402 10th Avenue South, Birmingham, Alabama 35294-1241
Email: selinger@uab.edu

DOI: https://doi.org/10.1090/jams/942
Received by editor(s): September 19, 2017
Received by editor(s) in revised form: July 5, 2019
Published electronically: June 16, 2020
Additional Notes: The first author was supported in part by Simons Foundation grant at the IMS, DFG grant BA4197/6-1, and ERC grant “HOLOGRAM”
The second author thanks the NSF for their continuing support.
Article copyright: © Copyright 2020 American Mathematical Society