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Equilateral triangulations and the postcritical dynamics of meromorphic functions

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Abstract

We show that any dynamics on any planar set S discrete in some domain D can be realized by the postcritical dynamics of a function holomorphic in D, up to a small perturbation. A key step in the proof, and a result of independent interest, is that any planar domain D can be equilaterally triangulated with triangles whose diameters \(\rightarrow 0\) (at any prescribed rate) near \(\partial D\).

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Acknowledgements

The authors would like to thank the anonymous referee for their suggestions which led to an improved version of the manuscript.

Funding

The first author was partially supported by NSF Grant DMS 1906259 and the third author was partially supported by Simons Grant 581668.

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Authors and Affiliations

  1. Stony Brook University, Stony Brook, USA

    Christopher J. Bishop

  2. University of North Texas, Denton, USA

    Kirill Lazebnik & Mariusz Urbański

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  1. Christopher J. Bishop
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  2. Kirill Lazebnik
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Correspondence to Kirill Lazebnik.

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Bishop, C.J., Lazebnik, K. & Urbański, M. Equilateral triangulations and the postcritical dynamics of meromorphic functions. Math. Ann. 387, 1777–1818 (2023). https://doi.org/10.1007/s00208-022-02507-4

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