Radial Julia Dimensions of Exponential Maps

Definitions and mathematical statement

For a complex parameter $\lambda$, define the entire map \[ f_\lambda:{\mathbb{C}}\longrightarrow{\mathbb{C}},\qquad f_\lambda(z)=\lambda e^z. \] Write $\dim_H A$ for the Hausdorff dimension of a subset $A\subseteq{\mathbb{C}}$, using the Euclidean metric. The source uses $J_r(f_\lambda)$ for the radial Julia set, but supplies no definition of this set or of a hyperbolic parameter. Let $D\subseteq{\mathbb{C}}$ denote the intended parameter domain and let \[ \mathcal H_D=\{\lambda\in D:\lambda\text{ is hyperbolic in the intended sense}\}. \] Neither $D$ nor the intended hyperbolicity criterion is specified in the supplied record; these remain unresolved qualifications rather than additional assumptions.

The supplied problem asks to determine \[ \sup\left\{\dim_H J_r(f_\lambda):\lambda\in\mathcal H_D\right\}, \] and to determine whether $\lambda\mapsto\dim_H J_r(f_\lambda)$ is increasing on the real interval $(0,1/e)$. If "increasing" means nondecreasing, the second question is whether \[ \forall\lambda_1,\lambda_2\in(0,1/e),\qquad \lambda_1<\lambda_2\ \Longrightarrow\ \dim_H J_r(f_{\lambda_1})\leq\dim_H J_r(f_{\lambda_2}). \] If strict increase is intended, replace $\leq$ by $<$. The record does not distinguish these interpretations. Its wording treats the dimension function as defined throughout this real interval, but supplies no explanation connecting that interval to the intended hyperbolic parameter set.

Unresolved source qualification. The supplied statement concerns exponential maps and radial Julia dimensions, whereas the supplied background and literature assessment describe an extraction ending with lattice gas Banach spaces and omitting the actual question. These incompatible descriptions prevent treating the statement's provenance as established. This is a conditional, statement-only draft requiring human mathematical and source review.

Short English statement

For the exponential family $f_\lambda(z)=\lambda e^z$, determine the supremum of radial Julia set dimensions over the intended hyperbolic parameters, and decide whether these dimensions increase as real $\lambda$ ranges over $(0,1/e)$. The parameter domain, relevant definitions, and strictness of increase require clarification. The supplied provenance assessment concerns a different mathematical setup.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30001962$, title Radial Julia Dimensions of Exponential Maps, source code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2012.
• Supplied report attribution: Mini-Workshop: Thermodynamic Formalism, Geometry and Stochastics (2012), Oberwolfach Reports, pp. 77--129, DOI 10.4171/owr/2012/03. Supplied source URL and literature-source link, labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2012/03. No problem-specific page or passage is supplied.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem open and states that the extraction ends at the setup of lattice gas Banach spaces, omits the actual question, and permits no verified mathematical resolution. The background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review confirming a self-contained, meaning-preserving cleanup without unsupported source-dependent additions. This is a supplied review claim, not independent verification, and does not resolve the conflicting literature assessment.