Components after removing a Cremer fixed point

Definitions and mathematical statement

Let $P_\alpha(z)=z^2+e^{2\pi i\alpha}z$, where $\alpha$ is irrational and the fixed point at zero is not locally holomorphically conjugate to multiplication by $e^{2\pi i\alpha}$. Such a fixed point is called a Cremer point. Let $K_\alpha=\{z:\sup_{n\ge0}|P_\alpha^n(z)|<\infty\}$ and $J_\alpha=\partial K_\alpha$ be the filled Julia set and Julia set. The latter is considered with its ordinary subspace topology in $\mathbb C$.

Remove just the fixed point and let \[ \mathcal C_\alpha=\{C:C\text{ is a maximal connected subset of } J_\alpha\setminus\{0\}\}. \] Determine the cardinality $|\mathcal C_\alpha|$ as $\alpha$ varies over quadratic Cremer parameters. In particular, the source asks whether this number is always countably infinite.

Connected components here mean maximal connected sets, not path components, complementary domains in the plane, or components of a small neighborhood of zero. These distinctions matter for a Julia set which is not locally connected. Countably infinite means a bijection with the positive integers; neither finite branching nor uncountably many components satisfies that proposal.

A satisfactory answer may prove a universal value, give counterexamples to it, or identify dynamical or arithmetic conditions under which the number differs. The operation is deletion of one point from the entire Julia set. Deleting the entire orbit of the critical point, taking a quotient model, or replacing components by externally accessible branches would ask different questions.

Short English statement

How many connected components remain when the Cremer fixed point is removed from a quadratic Cremer Julia set?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5300031 (AMR-052-0031), 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 P6. Original source indexed by AMR-052-0031. https://www.math.stonybrook.edu/open-problems-dynamical-systems.
• S3 Milnor, Problems on local connectivity, Problem6, in Bielefeld–Lyubich (1992). https://www.math.stonybrook.edu/preprints/ims92-7.pdf.