The distribution of geodesics on the cube surface

Abstract

We establish a Kronecker–Weyl type result, on time-quantitative equidistribution for a natural non-integrable system, geodesic flow on the cube surface. Our tool is the shortline-ancestor method developed in Beck, Donders, and Yang [Acta Math. Hungar. 161 (2020), 66–184] and Beck, Donders, and Yang [Acta Math. Hungar. 162 (2020), 220–324], modified in an appropriate way to embrace all slopes. The method is further enhanced by the symmetry of the cube through the use of the irreducible representations of the symmetric group S₄ which makes the determination of the irregularity exponent substantially simpler.