Josh Southerland

Collaborators

• Junaid Hasan
• Luke Jeffreys
• Chris Judge
• Dami Lee
• Tariq Osman
• Farbod Shokrieh
• Jane Wang
• Feng Zhu

Preprints

1. Affine mappings of translation surfaces: effective weak mixing (with Chris Judge)

abstract | arxiv | in preparation

2. Affine mappings of translation surfaces: shrinking targets and Diophantine properties (with Chris Judge)

abstract | arxiv | submitted

Let \((X,\omega)\) be a translation surface whose Veech group \(\Gamma\) is a lattice. We prove that the generic orbit of the group of affine homeomorphisms of \((X,\omega)\) 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 \(SL_2(\mathbb{R})/\Gamma\) 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 \(Y\).

3. Cylinder decompositions on geometric armadillo tails (with Dami Lee)

abstract | arxiv | submitted

We study a particular type 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.

4. Superdensity and bounded geodesics in moduli space

abstract | arxiv | under revisions

Following Beck-Chen, we say a flow \(\phi_t\) 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 \(\{\phi_t x\}_{0 \le t \le cT}\) 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.

Published

1. An effective slope gap distribution for lattice surfaces (with Tariq Osman and Jane Wang)

Discrete and Continuous Dynamical Systems, 2025

abstract | DOI | arxiv | cite

We prove an effective slope gap distribution result for translation surfaces whose Veech group is a lattice. As a corollary, we obtain a dynamical proof for an effective gap distribution result for the Farey fractions. As an intermediate step, we prove an effective equidistribution result for the intersection points of long horocycles with a particular transversal of the horocycle flow in \(SL_2(\mathbb{R})/\Gamma\) where \(\Gamma\) is a lattice.

Tex:

T. Osman, J. Southerland and J. Wang, An effective slope gap distribution for lattice surfaces, Discrete Contin. Dyn. Syst. {\bf 45} (2025), no.~12, 4998--5035; MR4938650

BibTex:

@article {MR4938650,
AUTHOR = {Osman, Tariq and Southerland, Josh and Wang, Jane},
TITLE = {An effective slope gap distribution for lattice surfaces},
JOURNAL = {Discrete Contin. Dyn. Syst.},
FJOURNAL = {Discrete and Continuous Dynamical Systems},
VOLUME = {45},
YEAR = {2025},
NUMBER = {12},
PAGES = {4998--5035},
ISSN = {1078-0947,1553-5231},
MRCLASS = {37A17 (32G15 37D40)},
MRNUMBER = {4938650},
MRREVIEWER = {Shucheng\ Yu},
DOI = {10.3934/dcds.2025081},
URL = {https://doi.org/10.3934/dcds.2025081},
}

2. Shrinking targets on square-tiled surfaces

New York Journal of Mathematics, 2024

abstract | link | arxiv | cite

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.

Tex:

J. Southerland, Shrinking targets on square-tiled surfaces, New York J. Math. {\bf 30} (2024), 656--681; MR4748465

BibTex:

@article {MR4748465,
AUTHOR = {Southerland, Josh},
TITLE = {Shrinking targets on square-tiled surfaces},
JOURNAL = {New York J. Math.},
FJOURNAL = {New York Journal of Mathematics},
VOLUME = {30},
YEAR = {2024},
PAGES = {656--681},
ISSN = {1076-9803},
MRCLASS = {37D40 (11J25 22E40 22F50 30F30 32G15 37A99)},
MRNUMBER = {4748465},
}

PhD Thesis / Masters Thesis

1. Quantitative density statements for translation surfaces, (Doctoral Thesis) pdf
2. The Laplacian: An Exploration and Historical Survey Tailored for Translation Surfaces, (Master's Thesis) pdf

Back to Home Page