Continuity of polynomial intertwining

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. Let $P_1$ be monic with connected Julia set and a repelling fixed point $x_0$. Choose an external ray landing at $x_0$ whose forward cycle consists of $q$ rays with combinatorial rotation number $p/q$: the dynamics permutes the cyclically ordered rays by a shift of $p$ positions. These rays cut the first dynamical plane into $q$ wedges. Let $P_2$ be monic with a ray of the same combinatorial rotation number landing at a repelling periodic point whose period divides $q$. Slit the second plane along the corresponding rays and insert the wedges from the first plane in the corresponding order. On the glued surface use $P_1$ and $P_2$ away from the cuts, adjusting the dynamics near inverse images of the cut rays to obtain continuity. This is the intertwining construction. The second landing orbit is required to be repelling; the source explicitly distinguishes the parabolic case, where this construction does not seem to work. In questions about variation, polynomial coefficients carry their usual topology; on a fixed-degree space this is equivalent to locally uniform convergence. Rational maps are considered up to conformal conjugacy, with the topology inherited from their fixed-degree coefficient space. The identifications used in the surgery are kept as part of the construction when the inputs vary.

Statement recorded in the supplied source.

With $P_1$ and the chosen ray-gluing data fixed, does the resulting polynomial realization vary continuously with $P_2$? [S2]

Short English statement

Does intertwining with a fixed first polynomial depend continuously on the second polynomial?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300011 (AMR-052-0011). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Bielefeld, Questions in Quasiconformal Surgery, in Ben Bielefeld and Mikhail Lyubich (editors), Problems in holomorphic dynamics (1992). Intertwining definition and Question 11, pp. 4–5. https://arxiv.org/pdf/math/9205209.