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Quasisymmetric embedding of self similar surfaces and origami with rational maps. (English) Zbl 1017.30027

The author uses an elementary and constructive method to show how self similar surfaces can be quasisymmetrically embedded in the plane. He builds a rational map which realizes the self similarity and for which every critical value is a repelling fixed point. Several examples of rational maps which realize subdivision rules are presented. How the Xmas tree, an embedded surface in \(\mathbb{R}^3\) whose Hausdorff dimension can be arbitrarily close to 3, can be embedded quasisymmetrically in the plane is also shown.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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