arXiv subjects

Grace Work

Publications and source records attributed to Grace Work.

5 recordsLinked to original sources

arXiv2024

Local product structure for equilibrium states of geodesic flows and applications

Let $S$ be a compact surface of genus $\geq 2$ equipped with a metric that is flat everywhere except at finitely many cone points with angles greater than $2\pi$. We examine the geodesic flow on $S$ and prove local product structure for a wide class of equilibrium states. Using this, we establish the Bernoulli property for these systems. We also establish local product structure for a similar class of equilibrium states for geodesic flows on rank 1, nonpositively curved manifolds.

math.DS↗

arXiv2021

Unique equilibrium states for geodesic flows on flat surfaces with singularities

Consider a compact surface of genus $\geq 2$ equipped with a metric that is flat everywhere except at finitely many cone points with angles greater than $2\pi$. Following the technique in the work of Burns, Climenhaga, Fisher, and Thompson, we prove that sufficiently regular potential functions have unique equilibrium states if the singular set does not support the full pressure. Moreover, we show that the pressure gap holds for any potential which is locally constant on a neighborhood of the singular set. Finally, we establish that the corresponding equilibrium states have the $K$-property, and closed regular geodesics equidistribute.

math.DS↗

arXiv2019

Discretely shrinking targets in moduli space

We consider the discrete shrinking target problem for Teichm\"uller geodesic flow on the moduli space of abelian or quadratic differentials and prove that the discrete geodesic trajectory of almost every differential will hit a shrinking family of targets infinitely often provided the measures of the targets are not summable. This result applies to any ergodic $\mathrm{SL}(2,\mathbb{R})$--invariant measure and any nested family of spherical targets. Under stronger conditions on the targets, we moreover prove that almost every differential will eventually always hit the targets. As an application, we obtain a logarithm law describing the rate at which generic discrete trajectories accumulate on a given point in moduli space. These results build on work of Kelmer and generalize theorems of Aimino, Nicol, and Todd.

math.DS↗

arXiv2016

A Transversal for horocycle flow on H(2)

Using zippered rectangle coordinates we parametrize a Poincar\'e section for horocycle flow on the space of genus 2 translation surfaces with one singular cone point of angle $6\pi$. In addition, we bound the return time under horocycle flow to this Poincar\'e section by examining a subset of surfaces where a certain sum of parameters is large.

math.GT↗

arXiv2014

The distribution of gaps for saddle connections on the octagon

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 surface has multiple cusps. We also show how to parametrize a Poincar\'e section for the horocycle flow on $SL(2,\mathbb{R})/SL(X,\omega)$ associated to an arbitrary Veech surface $(X, \omega)$. As a corollary, we show that the associated gap distribution is piecewise real analytic.

math.GT↗