Journal ArticleDOI
Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards
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TLDR
In this article, an asymptotic formula for the length spectrum of the billiard in isosceles triangles with angles (π/n, π /n,n−2/nπ) was given for the uniform distribution of infinite billiard trajectories in the same triangles.Abstract:
There exists a Teichmuller discΔ
n
containing the Riemann surface ofy
2+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 domain of finite (Poincare) volume inΔ
n
. Application is given to an asymptotic formula for the length spectrum of the billiard in isosceles triangles with angles (π/n, π/n,n−2/nπ) and to the uniform distribution of infinite billiard trajectories in the same triangles.read more
Citations
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Rational billiards and flat structures
Howard Masur,Serge Tabachnikov +1 more
TL;DR: In this paper, the authors show that rational billiards can be used to determine a flat surface in rational polygons and interval exchange maps in quadratic differentials in the unit square.
Journal ArticleDOI
Billiards and Teichmüller curves on Hilbert modular surfaces
TL;DR: In this paper, an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two, which lie on Hilbert modular surfaces parameterizing Abelian varieties with real multiplication.
Journal ArticleDOI
Deviation of ergodic averages for area-preserving flows on surfaces of higher genus
TL;DR: In this paper, a substantial part of a conjecture of Kontsevich and Zorich on the Lyapunov exponents of the Teichmuller geodesic flow on the deviation of ergodic averages for generic conservative flows on higher genus surfaces was proved.
Journal ArticleDOI
Affine mappings of translation surfaces: geometry and arithmetic
Eugene Gutkin,Chris Judge +1 more
TL;DR: In this article, the authors studied the relation between translation surfaces and quadratic differentials on a Riemann surface S. The main focus is the group Aff(S) and the associated group of differentials (S) ⊂ SL(2, R).
Journal ArticleDOI
Veech surfaces and complete periodicity in genus two
TL;DR: A classification of genus 2 Veech surfaces in the stratum with a single double zero is announced and all completely periodic translation surfaces in genus 2.
References
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Journal ArticleDOI
Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces With Applications to Dirichlet Series
TL;DR: The trace-formula as discussed by the authors is a general relation which can be considered as a generalization of the so-called Poisson summation formula (in one or more dimensions) and is used in many of these works.
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What is the Laplace Transform
TL;DR: The Laplace Transform as discussed by the authors is a Laplace transform that transforms the Laplace transformation into the Laplacian transform, and it can be seen as a form of transformation.