Hausdorff dimension of non-uniquely ergodic directions in eigenform loci

Yuming Wei; Pengyu Yang

Hausdorff dimension of non-uniquely ergodic directions in eigenform loci

Dynamical Systems 2026-07-27 v1

Authors: Yuming Wei , Pengyu Yang

Abstract

We consider the eigenform locus $\mathcal{E}_D$ in $\mathcal{H}(1,1)$ where $D$ is not a square. We prove that for any translation surface $(X,\omega) \in \mathcal{E}_D$ , the set of non-uniquely ergodic directions has Hausdorff dimension $1/2$ , except when $D=5$ and $(X,\omega)$ lies in the Teichm\"uller curve generated by the regular decagon.

Keywords

eigenform locus translation surface non-uniquely ergodic directions hausdorff dimension teichmuller curve regular decagon

Cite

@article{arxiv.2607.24181,
  title  = {Hausdorff dimension of non-uniquely ergodic directions in eigenform loci},
  author = {Yuming Wei and Pengyu Yang},
  journal= {arXiv preprint arXiv:2607.24181},
  year   = {2026}
}

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28 pages