arXiv subjects

Josh Southerland

Publications and source records attributed to Josh Southerland.

4 recordsLinked to original sources

arXiv2026

Affine mappings of translation surfaces: shrinking targets and Diophantine properties

Let $(X,ω)$ be a translation surface whose Veech group $Γ$ is a lattice. We prove that the generic orbit of the group of affine homeomorphisms of $(X,ω)$ can be used to approximate each point of $X$ with Diophantine precision. The proof utilizes an induced $SL_2(\mathbb{R})$-action on a fiber bundle $Y$ whose base is $SL_2(\mathbb{R})/Γ$ and whose fiber is $X$. We observe that this bundle embeds as an $SL_2(\mathbb{R})$-orbit closure in the moduli space of once marked translation surfaces, and hence we may invoke the spectral gap results of Avila and Gouëzel and a quantitative mean ergodic theorem for the $SL_2(\mathbb{R})$-action on the mean-zero, square-integrable functions on $Y$.

math.DS↗

arXiv2025

Cylinder decompositions on geometric armadillo tails

We study a class of finite-area, infinite-type translation surfaces, and find an explicit cylinder decomposition on these surfaces which do not manifest on finite-type translation surfaces. Each cylinder decomposition contains a special curve which we show is an obstruction to the existence of certain affine diffeomorphisms.

math.GT↗

arXiv2024

Superdensity and bounded geodesics in moduli space

Following Beck-Chen, we say a flow $ϕ_t$ on a metric space $(X, d)$ is superdense if there is a $c > 0$ such that for every $x \in X$, and every $T>0$, the trajectory $\{ϕ_t x\}_{0 \le t \le cT}$ is $1/T$-dense in $X$. We show that a linear flow on a translation surface is superdense if the associated Teichmüller geodesic is bounded. Conversely, if the linear flow is superdense, we show that along the Teichmüller geodesic, the diameter of the surface remains bounded. This generalizes work of Beck-Chen on lattice surfaces, and is reminiscent of work of Masur on unique ergodicity.

math.DS↗

arXiv2024

Shrinking targets on square-tiled surfaces

We study a shrinking target problem on square-tiled surfaces. We show that the action of a subgroup of the Veech group of a regular square-tiled surface exhibits Diophantine properties. This generalizes the work of Finkelshtein, who studied a similar problem on the flat torus.

math.DS↗