Hausdorff dimension of nonergodic directions in higher genus via effective equidistribution

Abstract

For each fixed $g\ge 3$, we prove that non-uniquely ergodic directions have Hausdorff dimension $1/2$ on almost every area-one surface in ${\mathcal{H}}^{\mathrm{odd}}(g-1,g-1)$ when $g$ is odd, and in ${\mathcal{H}}^{\mathrm{nonhyp}}(g-1,g-1)$ when $g$ is even. We construct these directions through cyclic splittings into $g$ slit tori. Effective equidistribution of Lindenstrauss--Mohammadi--Yang supplies sufficiently many children at each normal input. We estimate cusp and periodic-orbit failures with the original branching weights and prove that their total mass loss is summable. The surviving path measure yields the dimension lower bound, also for directions nonergodic for flat area.

arXiv comment25 pages