Invariant Analytic Jordan Curves of Rational Maps

Definitions and mathematical statement

Let $\widehat{{\mathbb{C}}}={\mathbb{C}}\cup\{\infty\}$ be the Riemann sphere. An analytic Jordan curve $\Gamma\subset\widehat{{\mathbb{C}}}$ is understood here to be the image of a real-analytic embedding of the circle into the sphere, with nonvanishing derivative in local coordinates. A rational self-map of the sphere is given by a rational function, interpreted at poles and at infinity.

For a specified class $\mathcal R$ of rational self-maps, define \[ \mathcal I(\mathcal R)=\left\{\Gamma\subset\widehat{{\mathbb{C}}}:\ \Gamma\text{ is an analytic Jordan curve and }\exists R\in\mathcal R\text{ with }R(\Gamma)=\Gamma\right\}. \] Under the convention that invariant means setwise invariant, the classification question in the supplied record is: characterize $\mathcal I(\mathcal R)$. Its additional question asks whether such invariant analytic Jordan curves can occur outside the examples described in the record as arising from rotation domains or Lattès functions.

Unresolved qualifications. The record does not specify the admissible degrees of $R$, so it does not determine $\mathcal R$. In particular, allowing the identity map makes every analytic Jordan curve invariant; a restriction such as $\deg R\geq2$ would be an additional hypothesis requiring confirmation. The record also does not specify whether invariance means $R(\Gamma)=\Gamma$, merely $R(\Gamma)\subseteq\Gamma$, or invariance under an iterate. The displayed formulation explicitly uses the first convention. Finally, the phrase "arising from rotation domains or Lattès functions" does not define a class of curves or pairs $(R,\Gamma)$, or the allowed constructions and equivalences. Consequently, the exceptional-example question cannot be given a definite quantified formulation from this record alone. The definition of analytic Jordan curve above is an explicit regularity convention that also requires confirmation against the intended source usage.

Short English statement

The supplied question asks which analytic simple closed curves on the Riemann sphere can be preserved by rational maps, and whether examples occur beyond those associated with rotation domains or Lattès functions. A precise final formulation requires the allowed map degrees, the meaning of invariance, and the meaning of those two families of examples. The supplied open label is a dated curation assessment, not an independently verified mathematical conclusion.

Sources

• Local UnsolvedMath registry: permanent ID $30001389$, source problem code catalogue entry, title Invariant Analytic Jordan Curves of Rational Maps; supplied from the project's registry records in data/unsolvedmath/problems.json and catalog/problems.json. This draft uses the supplied complete source record.
• Supplied report citation: Mini-Workshop: The Escaping Set in Transcendental Dynamics (2010), Oberwolfach Reports, pp. 2927--2964, DOI 10.4171/owr/2009/54. Supplied source URL and literature-source link, labeled "original OWR report": https://doi.org/10.4171/owr/2009/54. No specific problem page or passage is supplied.
• Attribution: the record gives proposed year 2009 and names no proposer. Its background attributes a historical proposal to Fatou, but supplies neither the quotation nor an exact location establishing that attribution for this formulation. The report title as supplied contains 2010, whereas the proposed year and DOI identifier contain 2009; these are retained without resolving their bibliographic relationship.
• Supplied literature assessment, checked exactly 2026-08-21: status "open." The assessment says that the record quotes Fatou's historical proposal, is incomplete, does not state a definite modern theorem, and that no resolution of the underlying open direction was identified. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified, with a supplied assertion 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 draft does not independently verify that assertion or reconcile it with the supplied assessment of incompleteness.