January  2022, 42(1): 319-326. doi: 10.3934/dcds.2021118

Subhyperbolic rational maps on boundaries of hyperbolic components

1. 

College of Mathematics and Statistics, Shenzhen University, Shenzhen 518061, China

2. 

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

3. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

* Corresponding author: Jinsong Zeng

Received  February 2021 Revised  June 2021 Published  January 2022 Early access  September 2021

In this paper, we prove that every quasiconformal deformation of a subhyperbolic rational map on the boundary of a hyperbolic component $ \mathcal{H} $ still lies on $ \partial \mathcal{H} $. As an application, we construct geometrically finite rational maps with buried critical points on the boundaries of some hyperbolic components.

Citation: Yan Gao, Luxian Yang, Jinsong Zeng. Subhyperbolic rational maps on boundaries of hyperbolic components. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 319-326. doi: 10.3934/dcds.2021118
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show all references

References:
[1]

B. Branner and N. Fagella, Quasiconformal Surgery in Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, 141, Cambridge University Press 2014.  Google Scholar

[2]

G. Cui and L. Tan, Hyperbolic-parabolic deformations of rational maps, Sci. China Math., 61 (2018), 2157-2220.  doi: 10.1007/s11425-018-9426-4.  Google Scholar

[3]

R. L. DevaneyN. FagellaA. Garijo and X. Jarque, Sierpiński curve Julia sets for quadratic rational maps, Ann. Acad. Sci. Fenn. Math., 39 (2014), 3-22.  doi: 10.5186/aasfm.2014.3903.  Google Scholar

[4]

H. Inou and S. Mukherjee, On the support of the bifurcation measure of cubic polynomials, Math. Ann., 378 (2020), 1-12.  doi: 10.1007/s00208-019-01826-3.  Google Scholar

[5]

K. Pilgrim and L. Tan, Spinning deformations of rational maps, Confor. Geom. Dyn., 8 (2004), 52-86.  doi: 10.1090/S1088-4173-04-00101-8.  Google Scholar

[6]

L. Tan, Stretching rays and their accumulations, following Pia Willumsen, Dynamics on the Riemann sphere, Eur. Math. Soc., Zürich, (2006), 183–208. doi: 10.4171/011-1/10.  Google Scholar

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