The closure of the space of Blaschke products in the Silverman and DeMarco compactified moduli spaces of rational maps, and realization of its boundary by polynomial dynamics

The space $B_d$ of degree-$d$ expanding Blaschke products embeds, via its natural marking, into a dynamical compactification of the Teichmüller space of expanding maps of the circle: the compactification is a sphere of invariant measures for $t\mapsto dt\bmod 1$, the boundary of the image is a sphere of dimension $2d-3$, and boundary points are pairs $(F,S)$ of a Blaschke product $F$ and a divisor of sources $S$ on the unit circle with $\deg F+\deg S=d$. The source paper explicitly leaves open the first of the two problems posed below, and documents the failure of polynomial realizability from which the second is developed. First, determine the closure of $B_d$ in the compactified moduli spaces of rational maps constructed by Silverman (geometric invariant theory) and by DeMarco (via barycenters of the measure of maximal entropy), comparing algebraic boundary data with the measure-theoretic boundary data of the dynamical compactification; only the closure in the $\mathbb{P}^{2d+1}$ compactification of the space of rational maps is settled. Second, characterize which boundary pairs are realized by polynomial dynamics: a monic polynomial $P$ with $P(0)=0$, locally connected filled Julia set and $0$ in an interior component determines a point of $\overline{B}_d$, but for $d=2$ realization can fail by the Cremer obstruction, and in degree $\ge3$ realizing polynomials are generally non-unique, with critical points outside the immediate basin contributing moduli. Later literature has produced major partial progress on the realization side — degeneration theory for Blaschke products and a complete boundary extension theorem for the Blaschke-product parameterization of the central hyperbolic component — but the algebraic closure question and the general realizability question with uniqueness remain open.

Background

An expanding map of the circle is a $C^1$ covering map $f:S^1\to S^1$ of degree $d>1$ whose derivative has modulus greater than one at every point; every such map is topologically conjugate to the model $p_d(t)=dt\bmod 1$, and a choice of conjugacy $\varphi$ is called a marking. The space of marked expanding maps of fixed degree is a Teichmüller-type space, and McMullen compactified it by allowing maps to degenerate to covering relations: multivalued maps of $S^1$ obtained as limits of graphs of genuine covering maps, so that points may be blown up to intervals. The resulting compactification is identified with the sphere of $p_d$-invariant probability measures $M_d(S^1)$ via $(f,\varphi)\mapsto\varphi_*(\lambda)$, where $\lambda$ is Lebesgue measure, in close analogy with the compactification of the Teichmüller space of a surface by measured geodesic laminations (equivalently, by geodesic currents). This is one instance of Sullivan's dictionary, the systematic collection of analogies between Kleinian groups and one-dimensional complex dynamics.

The holomorphic locus in this theory is the space $B_d$ of expanding Blaschke products: maps $f(z)=z\prod_{i=1}^{d-1}\frac{z-a_i}{1-\overline{a_i}z}$ with $|a_i|<1$, which are exactly the rational maps in the space above, holomorphic on the unit disk $\Delta$ with $0$ fixed; each preserves Lebesgue measure on $S^1=\partial\Delta$ and carries a natural marking, giving an inclusion $B_d\hookrightarrow T\mathrm{Exp}_d(S^1)$. Allowing the parameters $a_i$ to escape to the circle yields an algebraic compactification $\overline{B}_d\cong\overline{\Delta}^{(d-1)}$, the $(d-1)$-fold symmetric product of the closed disk, whose points are pairs $(F,S)$ consisting of a Blaschke product $F$ of degree $e\le d$ and an effective divisor $S=\sum m_i s_i$ of degree $d-e$ on $S^1$ (the sources, recording critical points that collide with the circle in the limit). The invariant-measure map embeds $\overline{B}_d$ into $M_d(S^1)$, the boundary of the image is a sphere of dimension $2d-3$, and on the stratum where $\deg F=e$ the support of the invariant measure $\nu_{(F,S)}$ is a set of Hausdorff dimension $\log e/\log d$, so the entropy drop of the degeneration is recorded measure-theoretically. Worked examples include the golden-mean Siegel disk polynomial $P(z)=\lambda z+z^2$, giving $(F,S)=(\lambda z,-\lambda)$ with $\nu$ supported on a dimension-zero Cantor set of external angles, and the superattracting cubic $P(z)=3z^2+4z^3$, giving $(z^2,-1)$ with dimension $\log 2/\log 3$.

On the algebraic side there are two natural compactifications of spaces of rational maps. Silverman compactified the moduli space $\mathrm{rat}_d=\mathrm{Rat}_d/\mathrm{PSL}_2(\mathbb{C})$ of degree-$d$ rational maps up to Möbius conjugacy using geometric invariant theory (GIT), obtaining a projective variety whose boundary represents degenerations of the dynamics. DeMarco compactified the space $\mathrm{Rat}_d$ itself inside $\mathbb{P}^{2d+1}$, where the limiting objects are algebraic correspondences — effective divisors of bidegree $(1,d)$ on $\mathbb{P}^1\times\mathbb{P}^1$ interpolating the limit dynamics — and in a second construction compactified the moduli space of quadratic rational maps using the barycenter of the measure of maximal entropy (the unique invariant measure of maximal Kolmogorov–Sinai entropy $\log d$). For the $\mathbb{P}^{2d+1}$ compactification, McMullen observed that the algebraic completion of $B_d$ agrees with its closure there, the pairs $(F,S)$ being exactly the limiting correspondences; for the compactified moduli spaces the corresponding closure question was left open.

The compactification also receives polynomial data. Let $P(z)=z^d+b_1z^{d-1}+\cdots+b_d$ be a monic polynomial with $P(0)=0$ and locally connected filled Julia set $K(P)$ — the set of points with bounded forward orbit, with Julia set $J(P)=\partial K(P)$; local connectivity is the regularity that makes the relevant Riemann maps extend continuously to the boundary, so that Julia-set points carry external-angle labels. Assume $0$ belongs to a component $U_0(P)$ of the interior of $K(P)$. Then $P$ determines a point of $\overline{B}_d$: the Riemann map of $(U_0(P),0)$ transports $P|_{U_0(P)}$ to a Blaschke product $F$ of degree $e$, and the canonical retraction $\rho:K(P)\to\overline{U_0(P)}$, which is locally constant outside its image, converts the $d-e$ critical points of $P$ outside $U_0(P)$ into a divisor $S$ of sources on $S^1$. For the resulting pair, the invariant measure $\nu_{(F,S)}$ is the harmonic measure (Brownian hitting measure from $0$) on $\partial U_0(P)$, and the measure of maximal entropy of $P$ pushes forward to the maximal-entropy measure of $(F,S)$. The converse fails in two documented ways: for $d=2$ a boundary point can be realized only by a polynomial conjugate to $\lambda z+z^2$ with $\lambda=f'(0)$, and when $\lambda\in S^1$ is very well approximated by roots of unity the fixed point $0$ is a Cremer point — an irrationally indifferent fixed point at which the map is not holomorphically linearizable — lying in the Julia set, so no basin component $U_0(P)$ exists; and for $d\ge3$ the realizing polynomial, when it exists, is generally not unique, the critical points outside $U_0(P)$ contributing additional moduli.

Problem Statement

The problem is to determine the boundary of the space $B_d$ of degree-$d$ expanding Blaschke products inside the algebraic compactifications of the moduli spaces of rational maps, and to characterize which boundary points are realized by polynomial dynamics. Two linked questions arising from McMullen's compactification theory for expanding maps on the circle — the first explicitly posed as an open question in the source paper, the second developed from the failure of realizability documented in its §4:

(Q1) Algebraic closure. Determine the closure of $B_d$, embedded via its natural marking in the space of marked expanding maps, in the two compactified moduli spaces of rational maps cited by the source: Silverman's geometric-invariant-theory compactification of the moduli space $\mathrm{rat}_d=\mathrm{Rat}_d/\mathrm{PSL}_2(\mathbb{C})$ in arbitrary degree, and DeMarco's compactification of the moduli space of quadratic rational maps via barycenters of the measure of maximal entropy. Identify which boundary points of these compactifications the Blaschke locus accumulates on, which of the pairs $(F,S)\in\overline{B}_d$ (a Blaschke product $F$ of degree $e$ together with a divisor of sources $S$ of degree $d-e$ on $S^1$) correspond to them, and how the algebraic boundary data compare with the measure-theoretic boundary data carried by the invariant measures $\nu_{(F,S)}$. The case of the $\mathbb{P}^{2d+1}$ compactification of $\mathrm{Rat}_d$ by algebraic correspondences is already settled in the source: the closure there agrees with the algebraic completion $\overline{B}_d$.

(Q2) Polynomial realization. Characterize the pairs $(F,S)\in\overline{B}_d$ that arise from a monic polynomial $P$ with $P(0)=0$, locally connected filled Julia set $K(P)$, and $0$ in a component $U_0(P)$ of the interior of $K(P)$, via the construction that transports $P|_{U_0(P)}$ to a Blaschke product and converts the critical points outside $U_0(P)$ into a divisor of sources. Among realizable pairs, determine the extent of uniqueness of the realizing polynomial, in particular the nature and dimension of the moduli contributed by the critical points of $P$ outside $U_0(P)$. The characterization must account for the two known obstructions: for $d=2$, realization forces $P$ to be conjugate to $\lambda z+z^2$ and fails by the Cremer obstruction when $\lambda\in S^1$ is very well approximated by roots of unity; for $d\ge3$, realizations are generally non-unique.

The common objective is a description of the boundary of the dynamical compactification of $B_d$ in algebro-geometric terms (Q1) and in concrete dynamical-systems terms (Q2); the two questions are linked because both ask which honest geometric objects — points of the compactified moduli spaces, and polynomials with prescribed basin dynamics — realize the measure-theoretic boundary data.

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