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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Pacman renormalization and self-similarity of the Mandelbrot set near Siegel parameters
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by Dzmitry Dudko, Mikhail Lyubich and Nikita Selinger HTML | PDF
J. Amer. Math. Soc. 33 (2020), 653-733 Request permission

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.
References
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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
  • MR Author ID: 969584
  • Email: dzmitry.dudko@stonybrook.edu
  • Mikhail Lyubich
  • Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794
  • MR Author ID: 189401
  • 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
  • MR Author ID: 874467
  • Email: selinger@uab.edu
  • 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.
  • © Copyright 2020 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 33 (2020), 653-733
  • MSC (2010): Primary 37E20, 37F25, 37F45
  • DOI: https://doi.org/10.1090/jams/942
  • MathSciNet review: 4127901