Elyn Rykken

Pseudo-Anosov homeomorphisms with quadratic expansion

We show that if f : M -M is a pseudo-Anosov homeomorphism on an orientable surface with oriented unstable manifolds and a quadratic expanding factor, then there is a hyperbolic toral automorphism on T2 and a map h : M -+ T2 such that h is a semi-conjugacy and (M, h) is a branched covering space of T2. We also give another characterization of pseudo-Anosov homeomorphisms with quadratic expansion in terms of the kinds of Euclidean foliations they admit which are compatible with the affine structure associated to f.

Constructions on the Sphere

8 April 2016 : : Math Horizons : : www.maa.org/mathhorizons Myles Dworkin and Elyn Rykken I n high school geometry, we learned various ruler and compass constructions—bisecting angles, drawing parallel lines, constructing regular polygons, and so on. Such constructions can be found in the most famous textbook in history, Euclid’s The Elements (c. 300 BCE). Surprisingly, the first four of the 13 chapters (or “books”) contain much of the plane geometry that is taught in high school today. In the 19th century, mathematicians discovered non-Euclidean geometries. These geometries have points, lines, and circles, but these objects may not behave in the ways their Euclidean counterparts do. In this article we look at one such geometry: spherical geometry. We investigate which Euclidean constructions still hold, which ones do not, and what new ones we obtain. (For a different approach to constructability on the sphere, see [3].)