Invariant Quasicircles for Rational Maps

Definitions and mathematical statement

For a rational self-map $R:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$, let $\operatorname{Crit}(R)$ be its critical set and put \[ P_R=\overline{\bigcup_{n\geq1}R^n(\operatorname{Crit}(R))} \subset\widehat{\mathbb{C}}. \] Since a quasicircle is closed, using the unclosed postcritical union gives the same containment requirement. A spherical quasicircle is the image of $S^1$ under a quasiconformal homeomorphism of $\widehat{\mathbb{C}}$; the source additionally requires $C\subset{\mathbb{C}}$, so its curve avoids infinity.

For a given rational map in the intended but unspecified class, the question is whether there exists such a planar quasicircle with \[ P_R\subseteq C, \qquad C\subseteq R^{-1}(C). \] The second condition is precisely $R(C)\subseteq C$, a forward-invariance containment; it is not the reverse containment or complete invariance. The usual nontrivial dynamical setting takes $\deg R\geq2$, but the source does not state a degree range.

The class of maps and coordinate convention are essential unresolved qualifications. A planar curve cannot contain a postcritical point at infinity. More generally a postcritical closure must fit inside a Jordan curve, and arbitrary rational maps are not assumed to satisfy this. The record does not impose postcritical finiteness, expansion, a Julia-set condition, or permission to replace $R$ by an iterate. These assumptions must not be silently imported from the literature discussion. Replacing the plane by the sphere or conjugating coordinates would also change the literal stated formulation and requires explicit justification.

The supplied literature triage dated 21 August 2026 reports quasicircle constructions in expanding or Julia-set-equals-the-sphere settings and later invariant finite graphs for postcritically finite rational maps. Its precise hypotheses and possible iterate conventions are not supplied. A finite invariant graph can branch and need not be a quasicircle, so the graph assessment alone does not answer the question. The curation calls the general quasicircle assertion unresolved, but that broad label does not supply the missing domain or remove the literal planar obstruction.

Short English statement

The requested curve must contain the postcritical closure and satisfy $R(C)\subseteq C$. The source leaves the map class and planar coordinate restrictions unspecified. Known graph constructions reported in the curation are not automatically invariant quasicircles, and possible iterate hypotheses need checking.

Sources

• Local UnsolvedMath registry, permanent ID $30000031$, exact source code catalogue entry, Oberwolfach Reports Open Problems. Statement status corrected_verified; supplied verification only converts the verbal containment to balanced notation.
• "Funktionentheorie (2004)," Oberwolfach Reports, pp. 349--406, DOI 10.4171/owr/2004/07: https://doi.org/10.4171/owr/2004/07. Proposed year 2004.
• Supplied report link: https://ems.press/content/serial-article-files/45883.
• Supplied "Invariant graphs of rational maps": https://arxiv.org/abs/1907.02870; supplied "Advances in Mathematics paper": https://doi.org/10.1016/j.aim.2022.108454. Complete author/theorem details are not supplied.
• Literature checked 21 August 2026: supplied status partially_solved. Registry created 25 August 2026 and updated 16 September 2026. The assessment is dated curation triage requiring independent verification before citation.
• Registry curation attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Underlying report and literature retain their supplied provenance; this statement-only draft requires human review.