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Transactions of the American Mathematical Society

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Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds

Author: Helena Mihaljević-Brandt
Journal: Trans. Amer. Math. Soc. 364 (2012), 4053-4083
MSC (2010): Primary 37F10; Secondary 30D05, 37F30, 37C15, 37D20
Published electronically: March 22, 2012
MathSciNet review: 2912445
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Abstract: 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

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.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.