Explicit Invariant Densities for Even Continued Fractions

Definitions and mathematical statement

For an irrational number $x\in[0,1]$, write its regular continued fraction as \[ x=[0;a_1(x),a_2(x),\ldots],\qquad a_n(x)\in\{1,2,3,\ldots\}. \] Define \[ E_{\mathrm{even}}=\{x\in[0,1]\setminus{\mathbb{Q}}: a_n(x)\in\{2,4,6,\ldots\}\text{ for every }n\geq1\}, \qquad h=\dim_{\mathrm H}E_{\mathrm{even}}. \] The Gauss map on irrational points of $(0,1)$ is \[ T(x)=\frac1x-\left\lfloor\frac1x\right\rfloor. \] It maps $E_{\mathrm{even}}$ into itself. Interpret the source's $P_h$ as the $h$-dimensional packing measure restricted to $E_{\mathrm{even}}$; its normalization is not specified. The source refers to a measure $\mu$ on this set satisfying \[ \mu(T^{-1}A)=\mu(A),\qquad \mu(A)=0\Longleftrightarrow P_h(A)=0 \] for every Borel set $A\subseteq E_{\mathrm{even}}$, where the inverse image is taken under the restricted map. The record does not specify a normalization or a uniqueness characterization of $\mu$, or supply the measure-theoretic hypotheses supporting the stated Radon--Nikodym derivative.

Recorded request, subject to unresolved qualifications. Find a closed explicit formula for the real-analytic extension of $\mathrm d\mu/\mathrm dP_h$ to $[0,1]$, or prove that no such formula exists. Under the usual neighborhood interpretation of real analyticity on a closed interval, the extension sought would be a real-analytic function $F:U\to{\mathbb{R}}$, for some open set $U\subseteq{\mathbb{R}}$ containing $[0,1]$, such that \[ F(x)=\frac{\mathrm d\mu}{\mathrm dP_h}(x) \quad\text{for }P_h\text{-almost every }x\in E_{\mathrm{even}}. \] This interpretation of endpoint analyticity requires confirmation. The wording presupposes such an extension but does not supply its existence or hypotheses ensuring it. Moreover, "closed explicit formula" has no specified class of admissible expressions, so its nonexistence alternative is not a precise mathematical assertion as supplied.

Source conflict. Although the statement fields give the request above, the supplied literature assessment says that the extracted text repeats the preceding problem and ends before the actual noise-induced phenomenon question. Consequently, the intended question associated with this permanent record cannot be securely identified from the supplied evidence. The displayed request is a qualified transcription, not a reconstruction of the missing question.

Short English statement

The recorded request concerns an analytic formula for the density of a Gauss-map invariant measure on numbers whose regular continued-fraction entries are all even. The measure normalization, endpoint analyticity convention, and meaning of "closed explicit formula" require clarification. The supplied curation also reports a truncated, duplicated extraction involving a different intended question, so this draft requires source and mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001958$, source problem code catalogue entry, titled "Explicit Invariant Densities for Even Continued Fractions," in the Oberwolfach Reports Open Problems collection. The record lists proposed year 2012 and supplies no proposer attribution.
• Supplied report citation: 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, labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2012/03. No passage-level citation establishing the displayed request is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "open" but reports that the extraction repeats the preceding problem and terminates before the actual noise-induced phenomenon question, leaving no complete claim to verify. This is supplied dated curation, not an independently verified literature assessment; its background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry separately labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review asserting a self-contained, meaning-preserving cleanup. These are supplied review labels, not independent confirmation of mathematical correctness or report fidelity, and they do not resolve the conflicting extraction assessment.