These surfaces arise in dynamical systems where they can be used to model billiards, and in Teichmüller theory.
be a collection of (not necessarily convex) polygons in the Euclidean plane and suppose that for every side
In the case where these two lines form a polygon (i.e. they do not intersect outside of their endpoints) there is a natural side-pairing.
the sum of the angles of the polygons around the vertices which map to it is a positive multiple of
Thus these differentials defined on chart domains glue together to give a well-defined holomorphic 1-form
The simplest example of a translation surface is obtained by gluing the opposite sides of a parallelogram.
-gon then the translation surface obtained by gluing opposite sides is of genus
The foliation is also obtained by the level lines of the imaginary part of a (local) primitive for
of translation surfaces which have a holomorphic form whose zeroes match the partition.
is the moduli space of tori, which is well-known to be an orbifold; in higher genus, the failure to be a manifold is even more dramatic).
This volume was proved to be finite independently by William A. Veech[1] and Howard Masur.
[2] In the 90's Maxim Kontsevich and Anton Zorich evaluated these volumes numerically by counting the lattice points of
From this observation they expected the existence of a formula expressing the volumes in terms of intersection numbers on moduli spaces of curves.
Alex Eskin and Andrei Okounkov gave the first algorithm to compute these volumes.
They showed that the generating series of these numbers are q-expansions of computable quasi-modular forms.
[3] More recently Chen, Möller, Sauvaget, and don Zagier showed that the volumes can be computed as intersection numbers on an algebraic compactification of
Currently the problem is still open to extend this formula to strata of half-translation surfaces.
Formally, a translation surface is defined geometrically by taking a collection of polygons in the Euclidean plane and identifying faces by maps of the form
The geometric structure obtained in this way is a flat metric outside of a finite number of singular points with cone angles positive multiples of
(which is invariant under half-translations), and for the other direction one takes the Riemannian metric induced by
, equivalently its action on the hyperbolic plane admits a fundamental domain of finite volume.
[11] A geodesic in a translation surface (or a half-translation surface) is a parametrised curve which is, outside of the singular points, locally the image of a straight line in Euclidean space parametrised by arclength.
Thus a maximal geodesic is a curve defined on a closed interval, which is the whole real line if it does not meet any singular point.
in the case of a half-translation surface) then the geodesics with direction theta are well-defined on
On a flat torus the geodesic flow in a given direction has the property that it is either periodic or ergodic.
In general this is not true: there may be directions in which the flow is minimal (meaning every orbit is dense in the surface) but not ergodic.
[12] On the other hand, on a compact translation surface the flow retains from the simplest case of the flat torus the property that it is ergodic in almost every direction.
[13] Another natural question is to establish asymptotic estimates for the number of closed geodesics or saddle connections of a given length.
This dynamical system is equivalent to the geodesic flow on a flat surface: just double the polygon along the edges and put a flat metric everywhere but at the vertices, which become singular points with cone angle twice the angle of the polygon at the corresponding vertex.
Then this map is an interval exchange transformation and it can be used to study the dynamic of the geodesic flow.