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Transactions of the American Mathematical Society

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

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds
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by Helena Mihaljević-Brandt PDF
Trans. Amer. Math. Soc. 364 (2012), 4053-4083 Request permission


Let $f:\mathbb {C}\rightarrow \mathbb {C}$ be a transcendental entire map that is subhyperbolic, i.e., the intersection of the Fatou set $\mathcal {F}(f)$ and the postsingular set $P(f)$ is compact and the intersection of the Julia set $\mathcal {J}(f)$ and $P(f)$ is finite. Assume that no asymptotic value of $f$ belongs to $\mathcal {J}(f)$ and that the local degree of $f$ at all points in $\mathcal {J}(f)$ is bounded by some finite constant. We prove that there is a hyperbolic map $g\in \{z\mapsto f(\lambda z):\; \lambda \in \mathbb {C}\}$ with connected Fatou set such that $f$ and $g$ are semiconjugate on their Julia sets. Furthermore, we show that this semiconjugacy is a conjugacy when restricted to the escaping set $I(g)$ of $g$. In the case where $f$ can be written as a finite composition of maps of finite order, our theorem, together with recent results on Julia sets of hyperbolic maps, implies that $\mathcal {J}(f)$ is a pinched Cantor bouquet, consisting of dynamic rays and their endpoints. Our result also seems to give the first complete description of topological dynamics of an entire transcendental map whose Julia set is the whole complex plane.
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Additional Information
  • Helena Mihaljević-Brandt
  • Affiliation: Mathematisches Seminar der Christian-Albrechts-Universität zu Kiel, 24118 Kiel, Germany
  • Email:
  • Received by editor(s): November 27, 2009
  • Received by editor(s) in revised form: July 5, 2010
  • Published electronically: March 22, 2012
  • Additional Notes: This work was supported by the Engineering and Physical Sciences Research Council (EPSRC), grant-code: EP/E05285, and was partly supported by the EU Research Training Network Cody.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 4053-4083
  • MSC (2010): Primary 37F10; Secondary 30D05, 37F30, 37C15, 37D20
  • DOI:
  • MathSciNet review: 2912445