Strong convergence, realization, and escape-rate control of geometric limits of degenerating Blaschke products

The space $\mathcal{B}_d$ of degree-$d$ Blaschke products fixing the origin admits two natural compactifications: an algebraic one whose boundary points are pairs $(F,S)$ (a lower-degree proper map plus a divisor of sources on the circle), and a geometric one by minimal branched coverings of ribbon $\mathbb{R}$-trees (McMullen, 2009). The source paper proved that every divergent sequence has a geometrically convergent subsequence, gave effective sufficient conditions for algebraic convergence to be strong (geometric limit equal to the suspension of the algebraic limit), classified the degree-two objects by rotation number and fan height, and showed that outside finitely many exceptional points the limiting length spectrum is read off from the tree. The open problem is to complete the comparison between the two compactifications: (1) give necessary and sufficient conditions, in terms of the algebraic data $(F,S)$ and the asymptotic escape and approach rates of the zeros, under which algebraic convergence is strong; (2) characterize which minimal ribbon $\mathbb{R}$-tree branched coverings of degree $d$ arise as geometric limits and parametrize the resulting boundary, extending the quadratic classification to $d\ge3$; (3) determine how the escape regime controls the geometric limit and the limiting length spectrum, including the exceptional sets.

Background

A proper holomorphic self-map of the unit disk $\Delta$ with a fixed point at the origin is, up to conjugacy, a Blaschke product $f(z)=z\prod_{i=1}^{d-1}\frac{z-a_i}{1-\overline{a_i}z},\qquad a_i\in\Delta,$ of degree $d\ge2$; the space of these maps is denoted $\mathcal{B}_d\cong\Delta^{(d-1)}$. Each $f\in\mathcal{B}_d$ extends to a measure-preserving covering map of the unit circle and carries a unique marking homeomorphism $\varphi:S^1\to S^1$ conjugating $f$ to the model map $p_d(z)=z^d$ with $\varphi=\mathrm{id}$ when $f=p_d$. Writing $L(C,f)=\log |(f^q)'(z)|$ for a periodic cycle $C$ of $p_d$ of period $q$ with $\varphi(z)\in C$, the values $L(C,f)$ form the marked length spectrum of $f$, which determines $f$; the underlying theme is that divergent sequences in $\mathcal{B}_d$ (sequences with no convergent subsequence in $\mathcal{B}_d$, equivalently sequences whose zeros approach the unit circle) develop expansion that is highly localized on the circle, at widely separated hyperbolic scales.

An $\mathbb{R}$-tree is a metric space in which any two points are joined by a unique arc isometric to an interval in $\mathbb{R}$; it is simplicial if it has a discrete vertex set off which it is a disjoint union of arcs. A ribbon structure is a cyclic ordering of the edges incident to each vertex, i.e. a planar embedding up to isotopy. A map between ribbon $\mathbb{R}$-trees is a branched covering if it is a local isometry outside finitely many critical points, near which it is modeled on $(r,\theta)\mapsto(r,n\theta)$; it is minimal if there is no proper totally invariant subtree containing the fixed point $p$. The connection with the disk is hyperbolic-geometric: under rescaling, the disk takes on the appearance of a tree, and the critical orbits and their convex hulls record the degeneration.

McMullen (2009) established the following framework. (i) A minimal branched covering of a ribbon $\mathbb{R}$-tree with a marked fixed point is automatically simplicial, has discrete orbits, and its quotient $T/\langle f\rangle$ is a finite tree (Theorem 1.1). (ii) Every divergent sequence $f_n\in\mathcal{B}_d$ has a subsequence converging geometrically, in the sense of Gromov-Hausdorff limits of rescaled convex hulls of critical orbits, to a degree-$d$ minimal branched covering of a ribbon $\mathbb{R}$-tree (Theorem 1.3). (iii) After normalization by $M(f)=\sup_z L_f(z)$, the length functions converge to a piecewise-constant limit $L(z)$ with at most $d-1$ laps, meaning $L(z)$ rises and falls at most $d-1$ times as $z$ traverses $S^1$; the limiting cycle lengths $L(C)$ span a finite-dimensional vector space over $\mathbb{Q}$ (Theorem 1.4, Corollary 1.5), and the values of the tree's translation-length function $L_f$ on the ends of $T$ lie in the subgroup of $\mathbb{R}$ generated by the edge lengths of $T/\langle f\rangle$ (Theorem 1.2). (iv) In degree two, the normalized minimal quadratic branched coverings are classified by the rotation number $\rho(f)\in\mathbb{R}/\mathbb{Z}$ together with, when $\rho=a/b$ is rational, the fan height $\delta(f)\in[-1,1]$, an invariant measuring the failure of $f^b(c)$ to land on the unique critical point $c$; the moduli space is a circle with each rational point blown up to an interval (Theorems 1.7 and 8.1). This parallels, under Sullivan's dictionary between rational maps and Kleinian groups, the Morgan-Shalen compactification of Teichmüller space by dual $\mathbb{R}$-trees; the polynomial counterpart of the construction had been given earlier by DeMarco and McMullen.

The space $\mathcal{B}_d$ also has an algebraic compactification $\overline{\mathcal{B}}_d\cong\overline{\Delta}^{(d-1)}$ obtained by taking the closure of the zero-divisor parametrization; a boundary point is a pair $(F,S)$ consisting of a proper map $F$ of degree strictly less than $d$ and a divisor of sources $S=\sum m_i s_i$ on the unit circle recording the escaped zeros. The suspension construction associates a ribbon-tree branched covering to each such pair, and the pair carries a marking relation $\Phi:S^1\to S^1$ that blows sources up to intervals; the escape interval $\Phi^*(s)=\{z\in\Phi(s): z^d\notin\Phi(F(s))\}$ labels points about to leave the influence of the source $s$. A sequence $f_n\to(F,S)$ converges strongly if its algebraic limit is $(F,S)$ and its geometric limit is the suspension of $(F,S)$; for strong limits the limiting spectrum is combinatorial: $L(C)=\sum_s |C\cap\Phi^*(s)|\in\mathbb{Z}$ up to finitely many exceptional points and cycles (Theorems 1.6 and 13.1). The paper proves effective sufficient conditions for strong convergence (Theorems 14.1 and 14.3): the sources are simple (multiplicity one), the zeros escape at comparable rates, meaning $(1-|a_n(i)|)^{1+\epsilon}=O_\epsilon(1-|a_n(j)|)$ for every $\epsilon>0$ and all pairs of escaping zeros $i,j$; and either no source is periodic and distinct sources have disjoint orbits under $F$, or the zeros approach their limits no worse than quadratically, meaning $|a_n(i)-b(i)|^{2+\epsilon}=O_\epsilon(1-|a_n(1)|)$ for every $\epsilon>0$, a condition satisfied by radial and by horocyclic approach. Without comparability or simplicity, or under approach tangential beyond the horocycle scale, the geometric limit need not be the suspension and the criteria do not apply; the paper does not give necessary conditions. Likewise, outside finitely many exceptional points the limiting length function agrees with the tree's normalized length function (Theorem 12.2), but those exceptional points are not classified in general degrees, and no effective procedure is given for computing the edge-length subgroup and the induced rational relations among limiting cycle lengths. In degree $d\ge3$ no parametrization of the minimal ribbon-tree coverings is known, and the source itself shows that a cubic tree can have arbitrarily many reversed edges (edges on which the covering reverses orientation) and arbitrarily many length-function values, so the degree-two model does not extend naively. The unresolved target is the completion of this comparison between the algebraic and the geometric compactification of $\mathcal{B}_d$.

Problem Statement

Common objective: complete the comparison between the algebraic compactification $\overline{\mathcal{B}}_d$ (boundary points $(F,S)$) and the geometric compactification of the space $\mathcal{B}_d$ of degree-$d$ Blaschke products by minimal branched coverings of ribbon $\mathbb{R}$-trees, in the sense of McMullen (2009): determine when the two limits agree, describe the set of possible geometric limits, and read off the limiting length spectra. The three deliverables below are the interlocking facets of this single comparison and are stated in the source paper's own terms.

(1) Strong convergence criterion. Let $f_n\in\mathcal{B}_d$ converge algebraically to $(F,S)\in\partial\mathcal{B}_d$, with zeros $a_n(i)\to b(i)$ where $|b(i)|=1$ for $i\le e$ and $|b(i)|<1$ otherwise. Give necessary and sufficient conditions, expressed in terms of the algebraic data $(F,S)$ and the asymptotic behavior of the zeros (their limits, the escape rates $1-|a_n(i)|$ and their comparability, and the approach rates $|a_n(i)-b(i)|$ relative to $1-|a_n(1)|$), under which $f_n$ converges strongly, i.e. the geometric limit is the suspension of $(F,S)$. The known sufficient conditions are: simple sources (each source of multiplicity one), comparable escape rates of all escaping zeros, i.e. $(1-|a_n(i)|)^{1+\epsilon}=O_\epsilon(1-|a_n(j)|)$ for every $\epsilon>0$ and all $i,j\le e$; and either there are no periodic sources and distinct sources have disjoint orbits under $F$, or the zeros approach their limits no worse than quadratically, i.e. $|a_n(i)-b(i)|^{2+\epsilon}=O_\epsilon(1-|a_n(1)|)$ for every $\epsilon>0$ and all $1\le i\le d-1$. Determine which hypotheses can be weakened or removed (in particular sources of multiplicity greater than one, and non-comparable escape rates) and which, if any, are necessary; exhibit the degenerations where strong convergence fails and describe their geometric limits.

(2) Realization and classification in degree $d$, especially $d\ge3$. Characterize the minimal branched coverings of ribbon $\mathbb{R}$-trees of degree $d$ that arise as geometric limits of sequences in $\mathcal{B}_d$, and determine the structure of the resulting boundary (the moduli space of normalized minimal degree-$d$ ribbon-tree coverings). In degree two the answer is the circle with rational points blown up, parametrized by rotation number and fan height; for $d\ge3$ no parametrization is known, and since cubic trees can have arbitrarily many reversed edges and length-function values, the degree-two model does not extend naively: the question, as posed in the successor literature (Luo, 2021), is whether there is a characterization of which branched coverings can arise as limits, together with a determination of the structure of the boundary.

(3) Escape regimes and limiting length spectra. Determine how the escape regime of the zeros controls (a) the geometric limit, beyond the regimes covered by the proved criteria, which include radial and horocyclic approach (in particular approach to the sources worse than quadratically, and non-comparable escape rates among distinct escaping zeros, where intermediate or degenerate tree limits may appear), and (b) the limiting normalized length function $L(z)$ and cycle lengths $L(C)$, including the classification of the finite exceptional sets where $L(z)$ fails to agree with the tree's length function $L_f(z)/M(f)$ (Theorem 12.2) or where the integer-valued cycle formula of Theorem 13.1 fails.

The target is the stated generality: arbitrary degree $d\ge2$ and arbitrary algebraic limits, and it is not confined to proper subclasses of degenerations (for example the quasi post-critically finite ones, i.e. degenerating sequences asymptotically controlled by finite critical-orbit data) or to particular degrees.

References

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