Minimizing entropy for translation surfaces

TL;DR

This paper investigates the entropy of specific translation surfaces within certain orbits, demonstrating that entropy is minimized at surfaces tiled by equilateral triangles with singularities at vertices.

Contribution

It establishes the minimal entropy configuration among translation surfaces in the $SL(2, eal)$ orbit of square-tiled surfaces, focusing on equilateral triangle tilings.

Findings

Entropy is minimized at equilateral triangle tiled surfaces.

Minimal entropy surfaces have singularities at vertices with a common cone angle.

The result applies to translation surfaces in the $SL(2, eal)$ orbit of square-tiled surfaces.

Abstract

In this note, we consider the entropy of unit area translation surfaces in the $SL(2, \mathbb R)$ orbits of square tiled surfaces that are the union of squares, where the singularities occur at the vertices and the singularities have a common cone angle. We show that the entropy over such orbits is minimized at those surfaces tiled by equilateral triangles where the singularities occur precisely at the vertices.