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We answer a question of Itai Benjamini by showing there is a \(K< \infty\) so that for any \(\epsilon >0\), there exist \(\epsilon\)-dense discrete sets in the hyperbolic disk that are homogeneous with respect to \(K\)-biLipschitz maps of the disk to itself. However, this is not true for \(K\) close to \(1\); in that case, every \(K\)-biLipschitz homogeneous discrete set must omit a disk of hyperbolic radius \(\epsilon(K)>0\). For \(K=1\), this is a consequence of the Margulis lemma for discrete groups of hyperbolic isometries.more » « less
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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
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We show that any PSLG has an acute conforming triangulation with an upper angle bound that is strictly less than 90 degrees and that depends only on the minimal angle occurring in the PSLG. In fact, all angles are inside the interval I_0= [theta_0, 90 -\theta_0/2] for some fixed theta_0>0 independent of the PSLG except for triangles T containing a vertex v where the PSLG has an interior angle theta_v < \theta_0; then T is an isosceles triangle with angles in I_v = [theta_v, 90 -\theta_v/2].more » « less
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This is a companion to the paper "Weil-Petersson curves, conformal energies, beta-numbers, and minimal surfaces". That paper gives various new geometric characterizations of Weil-Petersson in the plane that can be extended to curves in all finite dimensional Euclidean spaces. This paper deals with the 2-dimensional case, giving new proofs of some known characterizations, and giving new results for the conformal weldings of Weil-Petersson curves and a geometric characterization of these curves in terms of Peter Jones's beta-numbers.more » « less
