Abstract
Let \(f_*\) be a polynomial of degree d (\(\ge 3\)) with all critical points escaping to infinity except one periodic critical point with multiplicity \(d_*-1\). We show that \(f_*\) can be tuned by an arbitrary monic centered polynomial Q of degree \(d_*\) with connected Julia set. This generalizes a celebrate result on the existence of the baby Mandelbrot sets by Douady–Hubbard. In particular, if \(d=3\), we prove that \(\mathcal E(f_*)\) is homeomorphic to the Mandelbrot set, where \(\mathcal E(f_*)\) is the set of all the cubic polynomials that are externally conjugate to \(f_*\).
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Acknowledgements
The authors would like to express their gratitude to Mitsuhiro Shishikura for valuable suggestion. Hiroyuki Inou is partially supported by JSPS KAKENHI Grant Number JP18K03367.
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Inou, H., Wang, Y. Straightening Maps for Polynomials with One Bounded Critical Orbit. J Geom Anal 33, 314 (2023). https://doi.org/10.1007/s12220-023-01369-9
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DOI: https://doi.org/10.1007/s12220-023-01369-9