Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants

@article{Eskin2002ModuliSO,
  title={Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants},
  author={A. V. Eskin and Howard A. Masur and Anton Zorich},
  journal={Publications Math{\'e}matiques de l'Institut des Hautes {\'E}tudes Scientifiques},
  year={2002},
  volume={97},
  pages={61-179}
}
  • A. Eskin, H. Masur, A. Zorich
  • Published 14 February 2002
  • Mathematics
  • Publications Mathématiques de l'Institut des Hautes Études Scientifiques
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References

SHOWING 1-10 OF 39 REFERENCES
Connected components of the moduli spaces of Abelian differentials with prescribed singularities
Consider the moduli space of pairs (C,ω) where C is a smooth compact complex curve of a given genus and ω is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe
Siegel measures
The goals of this paper are first to describe and then to apply an ergodictheoretic generalization of the Siegel integral formula from the geometry of numbers. The general formula will be seen to
Quadratic differentials and foliations
This paper concerns the interplay between the complex structure of a Riemann surface and the essentially Euclidean geometry induced by a quadratic differential. One aspect of this geometry is the "
Moduli spaces of quadratic differentials
The cotangent bundle ofJ (g, n) is a union of complex analytic subvarieties, V(π), the level sets of the function “singularity pattern” of quadratic differentials. Each V(π) is endowed with a natural
Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials
We compute the asymptotics of the number of connected branched coverings of a torus as their degree goes to infinity and the ramification type stays fixed. These numbers are equal to the volumes of
Affine mappings of translation surfaces: geometry and arithmetic
1. Introduction. Translation surfaces naturally arise in the study of billiards in rational polygons (see [ZKa]). To any such polygon P , there corresponds a unique translation surface, S = S(P),
Square Tiled Surfaces and Teichmüller Volumes of the Moduli Spaces of Abelian Differentials
We present an approach for counting the Teichmuller volumes of the moduli spaces of Abelian differentials on a Riemann surface of genus g. We show that the volumes can be counted by means of counting
LYAPUNOV EXPONENTS AND HODGE THEORY
We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the
Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards
There exists a Teichmuller discΔ n containing the Riemann surface ofy2+x n =1, in the genus [n−1/2] Teichmuller space, such that the stabilizer ofΔ n in the mapping class group has a fundamental
Chapter 13 Rational billiards and flat structures
Publisher Summary The theory of mathematical billiards can be partitioned into three areas: convex billiards with smooth boundaries, billiards in polygons (and polyhedra), and dispersing and
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1
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