Paper folding, Riemann surfaces, and convergence of pseudo-Anosov sequences

Details Paper folding, Riemann surfaces, and convergence of pseudo-Anosov sequences Paper folding, Riemann surfaces, and convergence of pseudo-Anosov sequences

2010-10-17

arxiv.org/abs/1010.3448v1

Mathematics

Dynamical Systems

75 pages, 18 figures

Scientific paper

A method is presented for constructing closed surfaces out of Euclidean polygons with infinitely many segment identifications along the boundary. The metric on the quotient is identified. A sufficient condition is presented which guarantees that the Euclidean structure on the polygons induces a unique conformal structure on the quotient surface, making it into a closed Riemann surface. In this case, a modulus of continuity for uniformizing coordinates is found which depends only on the geometry of the polygons and on the identifications. An application is presented in which a uniform modulus of continuity is obtained for a family of pseudo-Anosov homeomorphisms, making it possible to prove that they converge to a Teichm\"uller mapping on the Riemann sphere.

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