Related papers: Oblivious points on translation surfaces
A translational surface is a tensor product surface constructed from two space curves by translating one along the other. These surfaces are common within geometric modeling and, since their description is parametric, it is desirable to…
A translation surface is a surface formed by identifying edges of a collection of polygons in the complex plane that are parallel and of equal length using only translations. We determined that the same circle packing can be realized on…
We investigated singular points of translation surfaces under the linearly independent condition. In this paper, as completion, we investigate singular points of translation surfaces under the linearly dependent condition, using the…
We construct an explicit family of finite-area, infinite-genus translation surfaces whose vertical translation flow is strongly mixing. This provides a positive answer to a question posed by Lindsey and Trevi\~no~\cite{LT}
A translation structure on a surface is an atlas of charts to the plane so that the transition functions are translations. We allow our surfaces to be non-compact and infinite genus. We endow the space of all pointed surfaces equipped with…
There are only a few invariants one classically associates with precompact translation surfaces, among them certain numberfields, i.e. fields which are finite extensions of the field Q of rational numbers. These fields are closely related…
Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as Abelian differentials on Riemann surfaces, and have moduli spaces…
We show that on any translation surface, if a regular point is contained in a simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in the unit circle. Moreover, the set of…
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 study connection points on the double regular $n$-gon translation surface, for $n \geq 7$ odd and its staircase model. For $n \neq 9$, we provide a large family of points with coordinates in the trace field that are not connection…
We identify all translation covers among triangular billiard surfaces. Our main tools are the holonomy field of Kenyon and Smillie and a geometric property of translation surfaces, which we call the fingerprint of a point, that is preserved…
A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface…
We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the…
We consider straight line flows on a translation surface that are minimal but not uniquely ergodic. We give bounds for the number of generic invariant probability measures.
We consider flow directions on the translation surfaces formed from double $(2n+1)$-gons, and give a sufficient condition in terms of a natural gcd algorithm for a direction to be hyperbolic in the sense that it is the fixed direction for…
A surface that is the pointwise sum of circles in Euclidean space is either coplanar or contains no more than 2 circles through a general point. A surface that is the pointwise product of circles in the unit-quaternions contains either 2,…
A fake octagon is a genus two translation surface with only one singular point and the same periods as the octagon. Existence of infinitely many fakes was first established by McMullen in 2007, and more generally follows from dynamical…
Consider a collection of finitely many polygons in $\mathbb C$, such that for each side of each polygon, there exists another side of some polygon in the collection (possibly the same) that is parallel and of equal length. A translation…
We consider how quickly a typical point returns to neighborhoods of itself under the flow in a typical direction on a translation surface.
Hallucinations in machine translation are translations that contain information completely unrelated to the input. Omissions are translations that do not include some of the input information. While both cases tend to be catastrophic errors…