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Free, publicly-accessible full text available February 1, 2026
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We strengthen the classical approximation theorems of Weierstrass, Runge, and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $$f$$ on a compact set $$K$$, the critical points of our approximants may be taken to lie in any given domain containing $$K$$, and all the critical values in any given neighborhood of the polynomially convex hull of $f(K)$.more » « less
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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 tend to 0 at any prescribed rate near the boundary. When D is the whole plane, the dynamical result was proved in "Prescribing the Postsingular Dynamics of Meromorphic Functions", by Bishop and Lazebnik by a different method (QC folding).more » « less
