Equilateral Triangulations and The Postcritical Dynamics of Meromorphic Functions

TL;DR

This paper demonstrates that complex planar dynamics can be realized through postcritical dynamics of holomorphic functions, utilizing a novel triangulation method of domains with shrinking triangles near boundaries.

Contribution

It introduces a method to realize arbitrary planar dynamics via holomorphic functions' postcritical sets, using equilateral triangulations with controlled shrinking near boundaries.

Findings

Any planar discrete set dynamics can be realized by holomorphic functions.

Domains can be triangulated with triangles shrinking to zero near the boundary.

The triangulation method is of independent interest for complex analysis.

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 $\rightarrow0$ (at any prescribed rate) near $\partial D$.