Combinatorial boundary of Blaschke-product space

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 $n>1$. A finite Blaschke product of degree $n$ is a proper holomorphic map of the unit disk $\Delta=\{z\in\mathbb C:|z|<1\}$ to itself of that degree. Let $A,B$ be such maps with $A(0)=B(0)=0$. Choose boundary fixed points $a,b$; there is a unique quasisymmetric conjugacy $h$ of the boundary circles with $h(a)=b$. Glue the disks by $h$ and transport the two dynamics to obtain a rational map $f(A,B)$ of the Riemann sphere, defined up to conformal conjugacy. Its Julia set is a quasicircle and the maps on the two complementary disks are conformally conjugate to $A$ and $B$. A quasicircle is a quasiconformal image of a circle. As in the source, the spaces below represent conformal-conjugacy classes; the boundary fixed-point choices are kept consistent, and additional finite combinatorial markings are suppressed. Write $\mathcal B(z^n)$ for the component of the degree-$n$ polynomial space containing $z^n$ in which there is an attracting fixed point whose immediate basin contains every finite critical point. Equivalently it is the space of maps $f(z^n,B)$. For fixed $A$, put $\mathcal B(A)=\{f(A,B):B\text{ varies}\}$. These spaces inherit complex structures from the polynomial or rational-map parameter space. Their actual boundaries are taken by closure in those degree-$n$ map spaces, with the same conformal-conjugacy convention. The natural biholomorphism is $F(f(z^n,B))=f(A,B)$. On the actual boundary of $\mathcal B(A)$ identify $f$ and $g$ when they are quasiconformally conjugate, meaning $u\circ f=g\circ u$ for a quasiconformal homeomorphism $u$ of the sphere. Write $\partial(A)$ for the resulting space with its quotient topology. The source equivalently describes these equivalence classes by connected complex submanifolds of the boundary; the quotient is non-Hausdorff for $n>2$.

Statement recorded in the supplied source.

Give a combinatorial description of the topological space $\partial(z^n)$. Such a description may use laminations, that is, noncrossing collections of chords or leaves encoding identifications, but the source does not require this particular form of model. [S2]

Short English statement

Describe the quotient boundary using combinatorial data, possibly laminations.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300016 (AMR-052-0016). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Curt McMullen, Rational maps and Teichmüller space, in Ben Bielefeld and Mikhail Lyubich (editors), Problems in holomorphic dynamics (1992). Problem following the quotient-boundary conjecture, p. 9. https://arxiv.org/pdf/math/9205209.