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arXiv:math/0609392math.DS

Flat Surfaces

Anton Zorich

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Abstract

Various problems of geometry, topology and dynamical systems on surfaces as well as some questions concerning one-dimensional dynamical systems lead to the study of closed surfaces endowed with a flat metric with several cone-type singularities. Such flat surfaces are naturally organized into families which appear to be isomorphic to the moduli spaces of holomorphic one-forms. One can obtain much information about the geometry and dynamics of an individual flat surface by studying both its orbit under the Teichmuller geodesic flow and under the linear group action. In particular, the Teichmuller geodesic flow plays the role of a time acceleration machine (renormalization procedure) which allows to study the asymptotic behavior of interval exchange transformations and of surface foliations. This long survey is an attempt to present some selected ideas, concepts and facts in Teichmuller dynamics in a playful way.

arXiv comments
(152 pages; 51 figures) Based on the lectures given by the author at the Les Houches School "Number Theory and Physics" in March of 2003 and at the workshop on dynamical systems in ICTP, Trieste, in July 2004. See "Frontiers in Number Theory, Physics and Geometry. Volume 1: On random matrices, zeta functions and dynamical systems'', P.Cartier; B.Julia; P.Moussa; P.Vanhove (Editors), Springer-Verlag (2006) for the entire collection (including, in particular, the complementary lectures of J.-C. Yoccoz). For a short version see the paper "Geodesics on Flat Surfaces", arXiv.math.GT/0609399
Journal reference
Frontiers in number theory, physics, and geometry. I, 437-583, Springer, Berlin, 2006
DOI
Subjects
math.DSDynamical Systems math.GTGeometric Topology

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  1. A Map of the World Won Japan’s Prestigious Design Award 1 comment

    • The only map I can think of from the sphere to the plane (and periodically tiles) uses genus 0 Riemann surfaces and quadratic differentials. And I know he didn't do that. Nonetheless, there's a fascinating work by Anton Zorich called "Flat Surfaces" (it's a great discussion with…

      mrcactu5View on HNParent context

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