Periodic orbits near a Cremer point

Definitions and mathematical statement

Let $R:\widehat{\mathbb C}\to\widehat{\mathbb C}$ be a rational map of degree at least two, and let $a$ be a fixed point with multiplier $R'(a)=e^{2\pi i\theta}$ for an irrational $\theta$. Derivatives at infinity are interpreted in a local coordinate. Assume $a$ is Cremer: no holomorphic coordinate near $a$ conjugates $R$ to its linear part.

Say that $a$ has the small-cycles property if every open neighborhood $U$ of $a$ contains infinitely many distinct finite periodic orbits of $R$. Each counted orbit \[ O(z)=\{z,R(z),\ldots,R^{q-1}(z)\},\qquad R^q(z)=z, \] must be contained wholly in $U$ and be different from the fixed orbit $\{a\}$. Does every Cremer fixed point of every rational map have this property?

For a periodic Cremer point of least period $r$, the corresponding formulation applies to the return map $R^r$ at that point. In terms of $R$, the small periodic cycles then lie near the full period-$r$ orbit, rather than near only one point on it.

Density of repelling periodic points in the Julia set is insufficient: it gives a periodic point near $a$ but does not keep its whole orbit there. The property is known for quadratic polynomials, while nonlinearizable holomorphic germs without small cycles exist. The question asks whether the requirement that the germ extend to a rational self-map rules out that local pathology in all degrees.

Short English statement

Does every Cremer point of a rational map have whole periodic cycles arbitrarily close to it?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5300033 (AMR-052-0033), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Stony Brook - Open Problems in Dynamical Systems, 1992, Milnor local P8. Original source indexed by AMR-052-0033. https://www.math.stonybrook.edu/open-problems-dynamical-systems.
• S3 Milnor, Problems on local connectivity, Problem8, in Bielefeld–Lyubich (1992). https://www.math.stonybrook.edu/preprints/ims92-7.pdf.