Entropy Production on Folded Hyperbolic Fractals

Definitions and mathematical statement

For a measurable dynamical system $(\Lambda,\mathcal S,f|_\Lambda)$, a measure $\mu$ on $(\Lambda,\mathcal S)$ is $f$-invariant if \[ \mu\left((f|_\Lambda)^{-1}(A)\right)=\mu(A) \qquad\text{for every }A\in\mathcal S. \] Here $\mathcal S$ denotes the measurable structure of the proposed construction; the supplied record does not specify that structure or require explicitly that the measures be probability measures.

The construction problem asks for further folded fractals $\Lambda$ that are hyperbolic for noninvertible maps $f$, together with families of $f$-invariant measures $\mu$ on $\Lambda$ having negative or positive entropy production. Writing $e_f(\mu)$ provisionally for the intended entropy production, the requested sign conditions are \[ e_f(\mu)<0 \qquad\text{or}\qquad e_f(\mu)>0. \] This notation names the requested quantity without supplying a definition absent from the record.

The statement is a construction request with unresolved specifications. The record does not define a folded fractal, specify the ambient space or regularity of $f$, give the intended notion of hyperbolicity for a noninvertible map, or define entropy production and the hypotheses under which it exists. It also does not identify the earlier examples relative to which the constructions must be "other," specify the size or parameterization of a family, or determine whether both signs must occur for one system, in separate systems, or whether either sign suffices. Consequently, the displayed inequalities describe the requested alternatives but do not resolve their quantifiers. These specifications must be recovered before this request can become a fully determined mathematical problem.

Short English statement

Construct further folded hyperbolic fractals for noninvertible maps, with families of invariant measures having strictly negative or strictly positive entropy production. The supplied record leaves essential definitions and the relationship between the two sign alternatives unresolved. Its partially solved label is supplied dated curation, not an independently verified mathematical conclusion.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30001957$, title "Entropy Production on Folded Hyperbolic Fractals," source problem code catalogue entry. The supplied record assigns the proposed year 2012 and names no individual proposer.
• Supplied source 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. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature-source link labeled "entropy production for folded systems": https://arxiv.org/abs/1104.2342. The record supplies no authors or further bibliographic details for this entry.
• Supplied literature assessment, checked on the exact date 2026-08-21: status "partially solved." The assessment states that the cited construction already supplies "negative inverse SRB entropy," that general entropy production formulas cover broad folded systems, and that finding further fractals and invariant families remains open. This is supplied dated curation derived from the OWR curation, not independently verified here; the record itself requests independent verification of time-sensitive status and cited claims.
• The supplied registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review describing the cleanup as meaning-preserving and self-contained. Those are supplied editorial assertions; they do not resolve the mathematical specifications missing from this record.