From hyperbolic to parabolic parameters along internal rays
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- by Yi-Chiuan Chen and Tomoki Kawahira;
- Trans. Amer. Math. Soc. 377 (2024), 4541-4583
- DOI: https://doi.org/10.1090/tran/9080
- Published electronically: May 15, 2024
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Abstract:
For the quadratic family $f_{c}(z) = z^2+c$ with $c$ in a hyperbolic component of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. In this paper we give a uniform derivative estimate of such a motion when the parameter $c$ converges to a parabolic parameter ${\hat {c}}$ radially; in other words, it stays within a bounded Poincaré distance from the internal ray that lands on ${\hat {c}}$. We also show that the motion of each point in the Julia set is uniformly one-sided Hölder continuous at ${\hat {c}}$ with exponent depending only on the petal number.
This paper is a parabolic counterpart of the authors’ paper “From Cantor to semi-hyperbolic parameters along external rays” [Trans. Amer. Math. Soc. 372 (2019), pp. 7959–7992].
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Bibliographic Information
- Yi-Chiuan Chen
- Affiliation: Institute of Mathematics, Academia Sinica, Taipei 106319, Taiwan
- MR Author ID: 725580
- Email: ycchen@math.sinica.edu.tw
- Tomoki Kawahira
- Affiliation: Graduate School of Economics, Hitotsubashi University, Tokyo 186-8602, Japan
- MR Author ID: 661650
- ORCID: 0000-0003-0329-3892
- Email: t.kawahira@r.hit-u.ac.jp
- Received by editor(s): April 22, 2023
- Published electronically: May 15, 2024
- Additional Notes: The first author was partly supported by MOST 108-2115-M-001-005 and 109-2115-M-001-006. The second author was partly supported by JSPS KAKENHI Grants numbers 16K05193 and 19K03535.
- © Copyright 2024 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 4541-4583
- MSC (2020): Primary 37F44; Secondary 37F46
- DOI: https://doi.org/10.1090/tran/9080
- MathSciNet review: 4778056