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Hausdorff dimension of biaccessible angles for quadratic polynomials

Volume 238 / 2017

Henk Bruin, Dierk Schleicher Fundamenta Mathematicae 238 (2017), 201-239 MSC: Primary 37F20; Secondary 37B10, 37E25, 37E45, 37F50. DOI: 10.4064/fm276-6-2016 Published online: 10 May 2017

Abstract

A point $c$ in the Mandelbrot set is called biaccessible if two parameter rays land at $c$. Similarly, a point $x$ in the Julia set of a polynomial $z \mapsto z^2+c$ is called biaccessible if two dynamic rays land at $x$. In both cases, we say that the external angles of these two rays are biaccessible as well.

We describe a purely combinatorial characterization of biaccessible (both dynamic and parameter) angles, and use it to give detailed estimates of the Hausdorff dimension of the set of biaccessible angles.

Authors

  • Henk BruinFaculty of Mathematics
    University of Vienna
    Oskar-Morgenstern-Platz 1
    1090 Wien, Austria
    e-mail
  • Dierk SchleicherJacobs University Bremen
    Research I
    P.O. Box 750 561
    D-28725 Bremen, Germany
    e-mail

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