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Geometry and Algebra of the Deltoid Map
Joshua P. Bowman
The geometry of the deltoid curve gives rise to a self-map of that is expressed in coordinates by . This is one in a family of maps th…
Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces
Joshua Bowman, Slade Sanderson
A homothety surface can be assembled from polygons by identifying their edges in pairs via homotheties, which are compositions of translation and scaling. We consider linear trajec…
Finiteness conditions on translation surfaces
Joshua P. Bowman
We consider various metric and analytic notions of finiteness on translation surfaces. The Veech group of a surface is discrete if the surface has finite area or is totally bounded…
Wild singularities of flat surfaces
Joshua P. Bowman, Ferrán Valdez
We consider flat surfaces and the points of their metric completions, particularly the singularities to which the flat structure of the surface does not extend. The local behavior…
The complete family of Arnoux-Yoccoz surfaces
Joshua P. Bowman
The family of translation surfaces constructed by Arnoux and Yoccoz from self-similar interval exchange maps encompasses one example from each genus greater than or…
Orientation-reversing involutions of the genus 3 Arnoux-Yoccoz surface and related surfaces
Joshua P. Bowman
We present a new description of the genus 3 Arnoux--Yoccoz translation surface in terms of its Delaunay polygons and show that, up to affine equivalence, it belongs to two families…