Continuous extension of external-ray rotation number

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Fix $n\geq2$ and let $\mathcal P_n$ consist of monic polynomials \[p(z)=z^n+a_{n-1}z^{n-1}+\cdots+a_1z,\qquad |a_1|\geq1,\] with the coefficient topology. An external ray is a ray approaching infinity at an angle $t\in\mathbb R/\mathbb Z$ in the Böttcher coordinate, continued towards the Julia set wherever possible. On external angles $p$ acts by $t\mapsto nt$. When at least one external ray lands at $0$, the rays landing there have a well-defined circular rotation number. Connectedness of $J(p)$ is not assumed.

Statement recorded in the supplied source.

Does this external-ray rotation number extend uniquely to a continuous map $\mathcal P_n\to\mathbb R/\mathbb Z$, taking the value $\theta$ whenever $a_1=e^{2\pi i\theta}$? [S2]

Short English statement

Does the rotation number of rays landing at the fixed point extend uniquely and continuously over the full stated polynomial space, with its prescribed value on unit-modulus multipliers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300074 (AMR-052-0074). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Bielefeld (editor) and conference contributors, Conformal Dynamics Problem List, arXiv:math/9201271v1 (1990), §2, Question 12 and preceding paragraph, printed p. 8. https://arxiv.org/abs/math/9201271v1.