# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2022, 42 (1) : 319-326. doi: 10.3934/dcds.2021118
##### References:
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##### References:
 [1] B. Branner and N. Fagella, Quasiconformal Surgery in Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, 141, Cambridge University Press 2014. [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. [3] R. L. Devaney, N. Fagella, A. 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. [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. [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. [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.
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