Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces

Abstract

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 trajectories on a \begin{document}$ 1 $\end{document} -parameter family of genus- \begin{document}$ 2 $\end{document} homothety surfaces. The closure of a trajectory on each of these surfaces always has Hausdorff dimension \begin{document}$ 1 $\end{document} , and contains either a closed loop or a lamination with Cantor cross-section. Trajectories have cutting sequences that are either eventually periodic or eventually Sturmian. Although no two of these surfaces are affinely equivalent, their linear trajectories can be related directly to those on the square torus, and thence to each other, by means of explicit functions. We also briefly examine two related families of surfaces and show that the above behaviors can be mixed; for instance, the closure of a linear trajectory can contain both a closed loop and a lamination.