Extremal Hausdorff Measures for Hyperbolic Exponential Maps

Definitions and mathematical statement

The supplied question concerns a hyperbolic exponential map $f(z)=\lambda e^z$. Under the usual convention for nonconstant complex exponential maps, $\lambda\in{\mathbb{C}}\setminus\{0\}$, and hyperbolicity means that the singular value $0$ is attracted to an attracting periodic orbit. These conventions clarify terminology; the record does not explicitly state the parameter domain or define hyperbolicity.

For real numbers $a<b$, write \[ S_{a,b}=\{z\in{\mathbb{C}}:a\leq\operatorname{Im}z\leq b\}. \] This is one possible boundary convention for a horizontal strip of finite width. The supplied record does not specify whether strip boundaries are included.

A Hausdorff gauge is a nondecreasing function $h:[0,\infty)\to[0,\infty)$ with $h(0)=0$ and $\lim_{r\downarrow0}h(r)=0$. With Euclidean diameter and an unnormalized convention, its Hausdorff measure is \[ H^h(E)=\lim_{\delta\downarrow0}\inf\left\{\sum_{j=1}^{\infty}h(\operatorname{diam}U_j):E\subseteq\bigcup_{j=1}^{\infty}U_j,\ \operatorname{diam}U_j\leq\delta\right\}. \] The record supplies neither the gauge $h$ nor the underlying set on which this measure is to be considered. In particular, it does not identify a Julia set, escaping set, or other carrier for the measure.

For clarity, one standard simultaneous Diophantine interpretation of extremality can be stated conditionally. Identify ${\mathbb{C}}$ with ${\mathbb{R}}^2$ by $z\mapsto(\operatorname{Re}z,\operatorname{Im}z)$, and, for $\varepsilon>0$, define \[ W_\varepsilon=\left\{x\in{\mathbb{R}}^2:\|qx-p\|_\infty<q^{-1/2-\varepsilon}\text{ for infinitely many }(p,q)\in{\mathbb{Z}}^2\times{\mathbb{Z}}_{\geq1}\right\}. \] In this interpretation, a Borel measure $\mu$ is extremal if $\mu(W_\varepsilon)=0$ for every $\varepsilon>0$.

If the intended gauge $h=h_f$, a Borel carrier $X_f\subseteq{\mathbb{C}}$, the preceding Diophantine interpretation, and the strip boundary convention are supplied, the question becomes: for every hyperbolic exponential map $f$ and every $a<b$, does the stipulated finite measure \[ \mu_{f,a,b}(E)=H^{h_f}(E\cap X_f\cap S_{a,b}),\qquad E\subseteq{\mathbb{C}}\text{ Borel}, \] satisfy \[ \mu_{f,a,b}(W_\varepsilon)=0\qquad\text{for every }\varepsilon>0, \] where $W_\varepsilon$ is transported to ${\mathbb{C}}$ by the stated identification? This is a conditional formulation, not a resolution of the missing source specifications. The word "finite" is retained as a supplied qualification; positivity is not stated.

There is also an unresolved mismatch within the record: its statement asks about extremality, whereas its background and literature assessment describe Problem 10 as asking for a gauge representing a conformal measure. The record does not define that conformal measure or explain how the two questions are related.

Short English statement

The supplied question asks whether the finite Hausdorff measures associated with a hyperbolic exponential map, restricted to every horizontal strip of finite width, are extremal in the sense attributed to Kleinbock, Lindenstrauss, and Weiss. A conditional standard interpretation is given above, but the gauge, measure carrier, strip boundaries, and exact Diophantine interpretation require clarification. The accompanying assessment instead concerns representation of a conformal measure, so the intended target also requires review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001398$, title: Extremal Hausdorff Measures for Hyperbolic Exponential Maps; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. No individual proposer is supplied; the proposed year is recorded as 2009.
• Supplied authoritative reference: Mini-Workshop: The Escaping Set in Transcendental Dynamics (2010), Oberwolfach Reports, pp. 2927--2964, DOI 10.4171/owr/2009/54. Source URL: https://doi.org/10.4171/owr/2009/54. The literature-sources entry labels this same link "original Oberwolfach problem report": https://doi.org/10.4171/owr/2009/54.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the report poses the existence of a gauge representing the conformal measure as Problem 10, with no later primary result located there establishing such a representation. This is supplied dated curation, not an independently verified assessment; it does not by itself establish the status of the extremality question.
• The supplied statement-status label is corrected_verified. Its accompanying verification note says that the mathematical content was checked against the official report and that notation, context, and question form were normalized without changing the source-level claim. This is a supplied verification claim, not verification performed for this draft.
• Extremality is attributed in the supplied statements to Kleinbock, Lindenstrauss, and Weiss; the original-statement field names Barak Weiss. No separate bibliographic reference defining their notion of extremality is supplied.