Continuity of polynomial mating

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 monic polynomials $P,Q$ of the same degree $d\geq2$, compactify each dynamical plane by a circle at infinity. On that circle the dynamics is $\theta\mapsto d\theta$. Glue the circles by identifying angle $\theta$ for $P$ with angle $-\theta$ for $Q$, and then collapse the external rays to points. The resulting quotient and induced map, when well defined, are the topological mating. The construction itself can fail to give a sphere or a rationally realizable map. 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.

Where mating has a rational realization, is that realization continuous when one input polynomial is fixed and the other varies continuously? Is mating jointly continuous as a function of both input polynomials? [S2]

Short English statement

Does mating depend continuously on each varying input, and is it a jointly continuous operation on pairs of polynomials?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300003 (AMR-052-0003). 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). Question 3, p. 3. https://arxiv.org/pdf/math/9205209.