Quasisymmetric uniformization and Hausdorff dimensions of Cantor circle Julia sets
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Abstract:
For Cantor circle Julia sets of hyperbolic rational maps, we prove that they are quasisymmetrically equivalent to standard Cantor circles (i.e., connected components are round circles). This gives a quasisymmetric uniformization of all Cantor circle Julia sets of hyperbolic rational maps.
By analyzing the combinatorial information of the rational maps whose Julia sets are Cantor circles, we give a computational formula of the number of the Cantor circle hyperbolic components in the moduli space of rational maps for any fixed degree.
We calculate the Hausdorff dimensions of the Julia sets which are Cantor circles, and prove that for any Cantor circle hyperbolic component $\mathcal {H}$ in the space of rational maps, the infimum of the Hausdorff dimensions of the Julia sets of the maps in $\mathcal {H}$ is equal to the conformal dimension of the Julia set of any representative $f_0\in \mathcal {H}$, and that the supremum of the Hausdorff dimensions is equal to $2$.
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Additional Information
- Weiyuan Qiu
- Affiliation: School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China
- Email: wyqiu@fudan.edu.cn
- Fei Yang
- Affiliation: Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China
- MR Author ID: 983714
- Email: yangfei@nju.edu.cn
- Received by editor(s): July 7, 2020
- Received by editor(s) in revised form: January 21, 2021
- Published electronically: April 27, 2021
- Additional Notes: This work was supported by National Natural Science Foundation of China (grant Nos. 11731003 and 12071210) and Natural Science Foundation of Jiangsu Province (grant No. BK20191246).
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 5191-5223
- MSC (2020): Primary 37F10; Secondary 37F20, 37F35
- DOI: https://doi.org/10.1090/tran/8392
- MathSciNet review: 4273190