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

TL;DR

This paper studies linear trajectories on genus-2 homothety surfaces, revealing their closures have Hausdorff dimension 1 and are either closed or lamination-like, with cutting sequences that are periodic or Sturmian, and relates these to trajectories on the square torus.

Contribution

It introduces a family of genus-2 homothety surfaces and explicitly relates their trajectories to those on the square torus, revealing new dynamical behaviors.

Findings

Trajectories have Hausdorff dimension 1 closures.

Closures contain closed loops or laminations with Cantor cross-sections.

Cutting sequences are either eventually periodic or Sturmian.

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 1-parameter family of genus-2 homothety surfaces. The closure of a trajectory on each of these surfaces always has Hausdorff dimension 1, 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.