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arXiv math/0609392math.DSmath.GT
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平坦表面

Flat Surfaces

Anton Zorich

备注：
(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
发表信息：
Frontiers in number theory, physics, and geometry. I, 437-583, Springer, Berlin, 2006
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摘要：

曲面上的几何、拓扑和动力系统的各种问题以及一维动力系统的一些问题导致了对具有多个锥型奇点的平坦度量的闭曲面的研究。这样的平坦曲面自然地被组织成与全纯单形式的模空间同构的族。通过研究一个平面在Teichmuller测地流和线性群作用下的轨道，可以得到关于它的几何和动力学的许多信息。特别是，Teichmuller测地流的时间加速机（重正化过程），它允许研究的渐近行为的间隔交换变换和表面叶理的作用。这个长的调查是试图提出一些选定的想法，概念和事实在Teichmuller动力学在一个有趣的方式。

英文摘要：

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.

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