Topological model for a Cremer Julia set

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 complex polynomial $P$ of degree at least two, its filled Julia set $K(P)$ is the set of points with bounded forward orbit, and $J(P)=\partial K(P)$ is its Julia set. An external ray is a radial curve in the coordinate near infinity that conjugates $P$ to its leading power map, continued into the basin of infinity. Its angle is measured in $\mathbb R/\mathbb Z$. A critical point is a zero of $P'$. Topological conjugacy means a homeomorphism $h$ satisfying $h\circ f=g\circ h$. A quasiconformal homeomorphism has bounded infinitesimal eccentricity; quasiconformal surgery cuts and glues dynamical planes and finds an invariant complex structure that can be straightened to a holomorphic map. For real irrational $\alpha$, put $P_\alpha(z)=z^2+\lambda z$, where $\lambda=e^{2\pi i\alpha}$. The fixed point zero is a Cremer point when no local holomorphic change of coordinate conjugates $P_\alpha$ to $z\mapsto\lambda z$ near zero. Such a $P_\alpha$ is called a Cremer polynomial. The request for a plausible topological model is the original qualitative modelling problem; it does not prescribe a lamination, a locally connected model, or a particular quotient construction.

Statement recorded in the supplied source.

Give a plausible topological model for the Julia set of a Cremer polynomial $P_\alpha$. [S2]

Short English statement

Describe a plausible topological model of the Julia set of a quadratic Cremer polynomial.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300027 (AMR-052-0027). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 John Milnor, Problems on local connectivity, in Ben Bielefeld and Mikhail Lyubich (editors), Problems in holomorphic dynamics (1992). Cremer-point definition and Problem 2, p. 21. https://arxiv.org/pdf/math/9205209.