## From Cantor to semi-hyperbolic parameters along external rays

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- by Yi-Chiuan Chen and Tomoki Kawahira PDF
- Trans. Amer. Math. Soc.
**372**(2019), 7959-7992 Request permission

## Abstract:

For the quadratic family $f_{c}(z) = z^2+c$ with $c$ in the exterior of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. Let $\hat {c}$ be a semi-hyperbolic parameter in the boundary of the Mandelbrot set. In this paper we prove that for each $z = z(c)$ in the Julia set, the derivative $dz(c)/dc$ is uniformly $O(1/\sqrt {|c-\hat {c}|})$ when $c$ belongs to a parameter ray that lands on $\hat {c}$. We also characterize the degeneration of the dynamics along the parameter ray.## References

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## Additional Information

**Yi-Chiuan Chen**- Affiliation: Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan
- MR Author ID: 725580
- Email: YCChen@math.sinica.edu.tw
**Tomoki Kawahira**- Affiliation: Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, Japan; and Mathematical Science Team, RIKEN Center for Advanced Intelligence Project (AIP), 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan
- MR Author ID: 661650
- Email: kawahira@math.titech.ac.jp
- Received by editor(s): March 8, 2018
- Received by editor(s) in revised form: December 24, 2018, and March 2, 2019
- Published electronically: June 17, 2019
- Additional Notes: The first author was partly supported by NSC 99-2115-M-001-007, MOST 103-2115-M-001-009, 104-2115-M-001-007, and 105-2115-M-001-003

The second author was partly supported by JSPS KAKENHI Grant Number 16K05193 - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 7959-7992 - MSC (2010): Primary 37F45; Secondary 37F99
- DOI: https://doi.org/10.1090/tran/7839
- MathSciNet review: 4029687