Related papers: Translation Covers Among Triangular Billiard Surfa…
In the present paper we study the $\SOL$ geometry that is one of the eight homogeneous Thurston 3-geomet\-ri\-es. We determine the equation of the translation-like bisector surface of any two points. We prove, that the isosceles property of…
Algebraically periodic directions on translation surfaces were introduced by Calta in her study of genus two translation surfaces. We say that a translation surface with three or more algebraically periodic directions is an algebraically…
The natural automorphism group of a translation surface is its group of translations. For finite translation surfaces of genus g > 1 the order of this group is naturally bounded in terms of g due to a Riemann-Hurwitz formula argument. In…
We show that every GL(2, R) orbit closure of translation surfaces is either a connected component of a stratum, the hyperelliptic locus, or consists entirely of surfaces whose Jacobians have extra endomorphisms. We use this result to give…
Given a Lagrangian submanifold in linear symplectic space, its tangent sweep is the union of its (affine) tangent spaces, and its tangent cluster is the result of parallel translating these spaces so that the foot point of each tangent…
We prove polynomial upper bounds for the deviation of ergodic averages for the straight line flow on every translation surface in almost every direction, in particular for those surfaces arising from rational polygonal billiards.
A translation surface in the three-dimensional sphere $\mathbb{S}^3$ is a surface generated by the quaternionic product of two curves, called generating curves. In this paper, we present rigidity results for such surfaces. We introduce an…
In this paper, we study and classify singular minimal translation surfaces in a Euclidean space of dimension 3 endowed with a certain semi-symmetric (non-)metric connection.
This paper gives sharp linear bounds on the genus of a normal surface in a triangulated compact, orientable 3--manifold in terms of the quadrilaterals in its cell decomposition---different bounds arise from varying hypotheses on the surface…
Total five different types of translation surfaces, based upon planarity of translating curves and the absolute figure, arise in a Galilean 3-space. Excepting the type in which both of translating curves are non-planar we obtain these…
We study the problem of covering R^d by overlapping translates of a convex body P, such that almost every point of R^d is covered exactly k times. Such a covering of Euclidean space by translations is called a k-tiling. The investigation of…
Given a pair of translation surfaces it is very difficult to determine whether they are supported on the same algebraic curve. In fact, there are very few examples of such pairs. In this note we present infinitely many examples of finite…
This is a survey about tiling by translation only and related questions and methods, especially those that have to do with Fourier Analysis.
We construct families of two-dimensional Sinai billiards whose transfer operators have Ruelle resonances arbitrarily close to 1. Our method involves taking a large enough cover of an initial billiard table, and relating the transfer…
Delaunay flip is an elegant, simple tool to convert a triangulation of a point set to its Delaunay triangulation. The technique has been researched extensively for full dimensional triangulations of point sets. However, an important case of…
We consider translation surfaces in the 3-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form $III$, i.e. their position vector $x$ satisfies the relation $\Delta^{III}x = \Lambda x$,…
This paper proves the following statement: If a convex body can form a three or fourfold translative tiling in the three-dimensional space, it must be a parallelohedron. In other words, it must be a parallelotope, a hexagonal prism, a…
A translation surface in the Heisenberg group $\mathrm{Nil}_3$ is a surface constructed by multiplying (using the group operation) two curves. We completely classify minimal translation surfaces in the Heisenberg group $\mathrm{Nil}_3$.
We prove that there exists a residual set of (non-rational) polygons such the billiard flow is weakly mixing with respect to the Liouville measure (on the unit tangent bundle to the billiard). This follows, via a Baire category argument,…
Let f = 0 be a hypersurface in n-dimensional affine space over a field k. We consider the pencil of hypersurfaces f- c = 0 with c varying over k.