External rays landing at a Cremer point

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.

For a polynomial $f$ of degree at least two, $f^n$ is its $n$-fold iterate, $K(f)=\{z\in\mathbb C:(f^n(z))_{n\geq0}\text{ is bounded}\}$ is its filled Julia set, and $J(f)=\partial K(f)$ is its Julia set. A space is locally connected if each point has a neighbourhood basis of connected sets. A Cremer periodic point $\zeta$ of period $p$ has $(f^p)'(\zeta)=e^{2\pi i\alpha}$ with $\alpha$ irrational and $f^p$ is not holomorphically linearizable at $\zeta$. An external ray is a trajectory from infinity of the gradient of the Green function $G_f(z)=\lim_{n\to\infty}(\deg f)^{-n}\log^+|f^n(z)|$ in the basin of infinity, labelled by its external angle. It lands at $\zeta$ if its endpoint limit on $J(f)$ is $\zeta$. For connected $K(f)$ these are the inverse images of radial rays under the Böttcher coordinate.

Statement recorded in the supplied source.

Can an external ray land at a Cremer point (including a Cremer periodic point)? [S2]

Short English statement

Can a ray approaching a polynomial Julia set from infinity end at a non-linearizable irrationally indifferent periodic point?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300029 (AMR-052-0029). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 John Milnor, "Problems on local connectivity", in Ben Bielefeld and Mikhail Lyubich (eds.), Problems in Holomorphic Dynamics (1992), Problem 4, pp. 21; arXiv:math/9205209v1. https://arxiv.org/pdf/math/9205209.