Intermediate Dimensions of Nonergodic Directions on Translation Surfaces

Definitions and mathematical statement

The supplied question concerns translation surfaces but does not specify compactness, genus, stratum, or the ambient space in which divergence is measured. The following gives a precise conditional formulation in the customary compact setting; these conventions require confirmation against the source.

Let $X=(S,\omega)$ be a compact connected Riemann surface with a nonzero holomorphic one-form, normalized so that its flat area measure $\mu_X$ satisfies $\mu_X(S)=1$. Away from the zeros of $\omega$, integration of $\omega$ supplies Euclidean coordinates whose transition maps are translations. Let $\Theta={\mathbb{R}}/\pi{\mathbb{Z}}$ be the space of unoriented directions, equipped with its angular metric. For $\theta\in\Theta$, let $\varphi_t^\theta$ denote straight-line flow in direction $\theta$, defined for almost every point for all times. Reversing the orientation does not change ergodicity.

Define the set of non-ergodic directions, with respect to flat area, by \[ N(X)=\left\{\theta\in\Theta:\ \exists\text{ a measurable }A\subseteq S,\quad 0<\mu_X(A)<1,\quad \mu_X\left((\varphi_t^\theta)^{-1}(A)\mathbin{\triangle}A\right)=0\ \text{for every }t\in{\mathbb{R}}\right\}. \] Here $\triangle$ denotes symmetric difference. This definition concerns failure of ergodicity of flat area; failure of unique ergodicity is a different possible intended meaning and is not substituted for it.

To formulate divergence, choose an ambient moduli space $\mathcal M$ containing the relevant area-one translation surfaces, with its usual topology. Write \[ g_t=\begin{pmatrix}e^t&0\\0&e^{-t}\end{pmatrix}, \] and let $r_{-\theta}$ rotate flat coordinates so that direction $\theta$ becomes horizontal. The matrices act by postcomposition on flat coordinate charts. Under this forward-ray convention, define \[ D_{\mathcal M}(X)=\left\{\theta\in\Theta:\ \text{for every compact }K\subseteq\mathcal M\text{ there exists }T\geq0\text{ such that }g_t r_{-\theta}X\notin K\text{ for all }t\geq T\right\}. \] The source record does not specify this convention or $\mathcal M$; in particular, divergence in a fixed stratum and divergence after projection to the moduli space of underlying Riemann surfaces must not be silently identified.

Question, under the stated conventions. Can either of the following existence statements hold? \[ \exists X\qquad 0<\dim_{\mathrm H}N(X)<\frac12, \] \[ \exists X\qquad 0<\dim_{\mathrm H}D_{\mathcal M}(X)<\frac12. \] Here Hausdorff dimension is computed in $\Theta$, and each existential quantifier ranges over the intended class of translation surfaces, which the supplied record leaves unspecified. These are separate alternatives: their witnessing surfaces need not coincide. This interpretation of the source's word "or" requires confirmation; the record does not explicitly ask for the dimension of a union or require realization of every value in $(0,1/2)$.

Short English statement

Can some translation surface have a set of non-ergodic directions, or a set of divergent directions, whose Hausdorff dimension lies strictly between $0$ and $1/2$? A precise final statement requires confirmation of the class of surfaces, the meaning of non-ergodicity, the ambient space and time convention for divergence, and the intended reading of "or."

Sources

• Local UnsolvedMath registry, permanent ID $30001783$, supplied complete source record; exact source problem code catalogue entry. Collection: Oberwolfach Reports Open Problems. The record gives proposed year 2011 and names no individual proposer.
• Billiards, Flat Surfaces, and Dynamics on Moduli Spaces (2011), Oberwolfach Reports, pp. 1361--1427, DOI 10.4171/owr/2011/25. Supplied source report and literature reference "original OWR report": https://doi.org/10.4171/owr/2011/25.
• Supplied literature reference "Cheung--Hubert--Masur dichotomy": https://arxiv.org/abs/0810.3956.
• Supplied literature reference "2026 extension": https://doi.org/10.1017/etds.2025.10265.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the Cheung--Hubert--Masur dichotomy and a 2026 extension cover important slit-surface families, while realization of an intermediate Hausdorff dimension is not settled in general. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is supplied verification metadata, not verification performed for this draft.