Related papers: The distribution of gaps for saddle connection dir…
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 compute the gap distribution of directions of saddle connections for two classes of translation surfaces. One class will be the translation surfaces arising from gluing two identical tori along a slit. These yield the first explicit…
We explicitly compute the limiting gap distribution for slopes of saddle connections on the flat surface associated to the regular octagon with opposite sides identified. This is the first such computation where the Veech group of the…
Translation surfaces with poles correspond to meromorphic differentials on compact Riemann surfaces. They appear in compactifications of strata of the moduli space of Abelian differentials and in the study of stability conditions. Such…
The slope gap distribution of a translation surface is a measure of how random the directions of the saddle connections on the surface are. It is known that Veech surfaces, a highly symmetric type of translation surface, have gap…
We explicitly compute the limiting slope gap distribution for saddle connections on any 2n-gon. Our calculations show that the slope gap distribution for a translation surface is not always unimodal, answering a question of Athreya. We also…
We consider saddle connections on a translation surface in a hyperelliptic connected component of a stratum that do not intersect the interior of a distinguished saddle connection. For this restricted set of saddle connections, we show that…
Flat surfaces that correspond to $k$-differentials on compact Riemann surfaces are of finite area provided there is no pole of order $k$ or higher. We denote by \textit{flat surfaces with poles of higher order} those surfaces with flat…
Dilation surfaces are generalizations of translation surfaces where the transition maps of the atlas are translations and homotheties with a positive ratio. In contrast with translation surfaces, the directional flow on dilation surfaces…
Fix a translation surface $X$, and consider the measures on $X$ coming from averaging the uniform measures on all the saddle connections of length at most $R$. Then as $R\to\infty$, the weak limit of these measures exists and is equal to…
We give an explicit formula for the limiting gap distribution of slopes of saddle connections on the golden L, or any translation surface in its SL(2, R)-orbit, in particular the double pentagon. This is the first explicit computation of…
In this paper, we study the distribution of renormalized gaps between slopes of saddle connections on translation surfaces. Specifically, we describe a procedure for finding the "winning holonomy vectors" as defined by…
For almost every flat surface the sequence of saddle connection lengths listed in increasing order is uniformly distributed mod one.
We describe typical degenerations of quadratic differentials thus describing ``generic cusps'' of the moduli space of meromorphic quadratic differentials with at most simple poles. The part of the boundary of the moduli space which does not…
In translation surfaces of finite area (corresponding to holomorphic differentials), directions of saddle connections are dense in the unit circle. On the contrary, saddle connections are fewer in translation surfaces with poles…
Configurations of rigid collections of saddle connections are connected component invariants for strata of the moduli space of quadratic differentials. They have been classified for strata of Abelian differentials by Eskin, Masur and…
We locate gaps in the spectrum of a Hamiltonian on a periodic cuboidal (and generally hyperrectangular) lattice graph with $\delta$ couplings in the vertices. We formulate sufficient conditions under which the number of gaps is finite. As…
We consider a flat metric with conical singularities on the sphere. Under the assumption that no partial sum of angle defects is equal to $2\pi$, we draw on the geometry of immersed disks to obtain an explicit upper bound on the number of…
For every flat surface, almost every flat surface in its $\mathsf{SL}(2,\mathbb{R})$ orbit has the following property: the sequence of its saddle connection lengths in non-decreasing order is uniformly distributed in the unit interval.
We provide a novel proof that the set of directions that admit a saddle connection on a meromorphic quadratic differential with at least one pole of order at least two is closed, which generalizes a result of Bridgeland and Smith, and…