Poincaré functions with spiders’ webs
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- by Helena Mihaljević-Brandt and Jörn Peter PDF
- Proc. Amer. Math. Soc. 140 (2012), 3193-3205 Request permission
Abstract:
For a polynomial $p$ with a repelling fixed point $z_0$, we consider Poincaré functions of $p$ at $z_0$, i.e. entire functions $\mathfrak {L}$ which satisfy $\mathfrak {L}(0)=z_0$ and $p(\mathfrak {L}(z))=\mathfrak {L}( p’(z_0) \cdot z)$ for all $z\in \mathbb {C}$. We show that if the component of the Julia set of $p$ that contains $z_0$ equals $\{z_0\}$, then the (fast) escaping set of $\mathfrak {L}$ is a spider’s web; in particular, it is connected. More precisely, we classify all linearizers of polynomials with regard to the spider’s web structure of the set of all points which escape faster than the iterates of the maximum modulus function at a sufficiently large point $R$.References
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Additional Information
- Helena Mihaljević-Brandt
- Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, 24118 Kiel, Germany
- Email: helenam@math.uni-kiel.de
- Jörn Peter
- Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, 24118 Kiel, Germany
- Email: peter@math.uni-kiel.de
- Received by editor(s): September 28, 2010
- Received by editor(s) in revised form: February 16, 2011, and March 28, 2011
- Published electronically: January 20, 2012
- Additional Notes: The second author has been supported by the Deutsche Forschungsgemeinschaft, Be 1508/7-1. He was also partially supported by the EU Research Training Network Cody.
- Communicated by: Mario Bonk
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 3193-3205
- MSC (2010): Primary 30D05; Secondary 37F10, 30D15, 37F45
- DOI: https://doi.org/10.1090/S0002-9939-2012-11164-5
- MathSciNet review: 2917092