Number of degenerate Herman rings

Definitions and mathematical statement

For a rational map $f$ of degree $d\geq2$, let $J(f)$ be its Julia set. Call a Jordan curve $C$ a degenerate Herman ring of $f$ if all the following conditions hold. The curve is contained in $J(f)$ and invariant under $f$. The restriction $f|_C$ is topologically conjugate to an irrational rotation: there are a homeomorphism $h:\mathbb R/\mathbb Z\to C$ and $\theta\notin\mathbb Q$ such that $f(h(t))=h(t+\theta)$. Finally, $C$ is neither a generalized circle nor a boundary component of a rotation domain.

A generalized circle means a round circle or an extended straight line. A rotation domain is a periodic Fatou component whose return map is analytically conjugate to an irrational rotation on a disk or annulus. No smoothness or analyticity is assumed for the Jordan curve $C$.

Let $\mathcal R(f)$ be the set of distinct curves satisfying these conditions, counted as subsets of the sphere. Is $\mathcal R(f)$ finite for every rational map $f$? More strongly, does there exist a finite integer $B(d)$ such that \[ \#\mathcal R(f)\leq B(d) \quad\text{whenever }\deg f=d? \] The first question permits a bound depending on the individual map; the second asks for a bound depending only on degree. The count concerns invariant degenerate rings themselves, not the ordinary annular Fatou components called Herman rings.

Short English statement

Are there finitely many degenerate Herman rings, and can their number be bounded using only the rational map's degree?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700008 (AMR-036-0008), 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 2. Original source indexed by AMR-036-0008. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Analytic invariant curves for rational functions, Question 2. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/invariant.pdf.