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Proceedings of the American Mathematical Society

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

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hausdorff dimension and biaccessibility for polynomial Julia sets
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by Philipp Meerkamp and Dierk Schleicher PDF
Proc. Amer. Math. Soc. 141 (2013), 533-542 Request permission

Abstract:

We investigate the set of biaccessible points for connected polynomial Julia sets of arbitrary degrees $d\geq 2$. We prove that the Hausdorff dimension of the set of external angles corresponding to biaccessible points is less than $1$, unless the Julia set is an interval. This strengthens theorems of Stanislav Smirnov and Anna Zdunik: they proved that the same set of external angles has zero $1$-dimensional measure.
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Additional Information
  • Philipp Meerkamp
  • Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853-4201
  • Email: pmeerkamp@math.cornell.edu
  • Dierk Schleicher
  • Affiliation: School of Engineering and Science, Jacobs University, Postfach 750 561, D-28725 Bremen, Germany
  • MR Author ID: 359328
  • Email: dierk@jacobs-university.de
  • Received by editor(s): April 14, 2011
  • Received by editor(s) in revised form: June 28, 2011
  • Published electronically: June 4, 2012
  • Communicated by: Bryna Kra
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 533-542
  • MSC (2010): Primary 37F10, 37F20, 37F35
  • DOI: https://doi.org/10.1090/S0002-9939-2012-11323-1
  • MathSciNet review: 2996957