Analytic degenerate Herman rings

Definitions and mathematical statement

Let $f$ be a rational map of degree at least two on the Riemann sphere. A Jordan curve is the image of a homeomorphism from the unit circle; it is analytic if it admits a real-analytic embedded parametrization with nonvanishing tangent. Write $J(f)$ for the Julia set, the complement of the region where the iterates form a normal family.

Does there exist such an $f$ and an analytic Jordan curve $C\subset J(f)$ with $f(C)=C$ for which a homeomorphism $h:\mathbb R/\mathbb Z\to C$ and an irrational real number $\theta$ satisfy \[ f(h(t))=h(t+\theta)? \] Require additionally that $C$ is not a generalized circle (a round circle or a line through infinity) and is not a boundary component of a rotation domain of $f$.

Here a rotation domain is a periodic Fatou component on which a return iterate is analytically conjugate to an irrational rotation of a disk or an annulus, respectively a Siegel disk or a Herman ring. Its usual interior rotation curves do not meet the Julia set. A curve satisfying the stated Julia-set and exclusion conditions is called a degenerate Herman ring. The regularity demanded is real analyticity, not merely smoothness or topological embeddedness. An example must verify the irrational rotation dynamics and the absence of both excluded geometries.

Short English statement

Can a rational map have a real-analytic degenerate Herman ring?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700007 (AMR-036-0007), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Analytic invariant curves, Question 1 (2025 update). Original source indexed by AMR-036-0007. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Analytic invariant curves for rational functions, Question 1. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/invariant.pdf.