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Hyperbolic-parabolic deformations of rational maps

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Abstract

We develop a Thurston-like theory to characterize geometrically finite rational maps, and then apply it to study pinching and plumbing deformations of rational maps. We show that under certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11125106).

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Authors and Affiliations

  1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Guizhen Cui

  2. LAREMA, Université d’Angers, Angers, 49045, France

    Lei Tan

Authors
  1. Guizhen Cui
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  2. Lei Tan
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Correspondence to Guizhen Cui.

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Dedicated to the Memory of Professor Lei Tan

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

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  • DOI: https://doi.org/10.1007/s11425-018-9426-4

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