McMullen's open questions on cycle lengths for Blaschke and quasi-Blaschke products: critical points of the length functional, uniqueness of the binding minimum, a uniform simplicity threshold, compactness, and an intersection number for cycles

This problem records the five open questions posed in McMullen's 2010 paper on dynamics of Blaschke products on the unit disk. For a Blaschke product $f$ and a cycle $C$ of the model map $p_d(z)=z^d$ on the unit circle, the cycle length $L(C,f)=\log|(f^q)'(z)|$ is computed at the periodic orbit of $f$ corresponding to $C$ under the canonical marking. The paper proves that short cycles (below $\log 2$) are simple, that every Blaschke product has a simple cycle of length $O(d)$, and that binding collections of cycles give compact sublevel sets in the moduli space of rational maps. It then asks: (Q1) whether $L(C,f)$ is free of critical points on the Blaschke space for simple $C$; (Q2) whether the binding sum $\sum_i L(C_i,f)$ achieves its minimum at a unique point; (Q3) whether a uniform simplicity threshold $\delta_d>0$ exists over the larger space $\mathrm{QB}_d$ of quasi-Blaschke products; (Q4) whether the compactness theorem for a binding pair of cycles extends from Blaschke products to $\mathrm{QB}_d$; and (Q5) whether there is a useful notion of intersection number for pairs of cycles. The hyperbolic-surface analogs of these statements are classical theorems.

Background

A Blaschke product of degree $d$ is a holomorphic self-map of the unit disk of the form $f(z)=z\prod_{i=1}^{d-1}\frac{z-a_i}{1-\overline{a_i}z}$ with $|a_i|<1$; the space of such maps is denoted $\mathcal{B}_d$ and is identified with the polydisk $\Delta^{d-1}$ via the zeros $a_i$. Each $f\in\mathcal{B}_d$ restricts to a covering map $S^1\to S^1$ of degree $d$; $f|_{S^1}$ preserves Lebesgue measure on the circle and is expanding there, with $|f'|>1$, so every periodic orbit (cycle) on $S^1$ has multiplier of modulus greater than 1, where the multiplier of a periodic point $z$ of period $q$ is the derivative $(f^q)'(z)$, the infinitesimal expansion rate of the $q$-th iterate at $z$. The model map is the monomial $p_d(z)=z^d$. The canonical marking $\phi_f:S^1\to S^1$ is the unique orientation-preserving homeomorphism that varies continuously with $f$, conjugates $f$ to $p_d$ on the circle, and equals the identity when $f=p_d$; it labels the periodic orbits of $f$ by the cycles of $p_d$. For a cycle $C$ of $p_d$ of period $q=|C|$, the cycle length is $L(C,f)=\log|(f^q)'(z)|$ for any $z\in S^1$ with $\phi_f(z)\in C$, i.e. for any point of the periodic orbit of $f$ marked by $C$; this is independent of the chosen point on that orbit. For a finite $p_d$-invariant set $X\subset S^1$, the degree $\deg(p_d|_X)$ is the least $e>0$ such that $p_d|_X$ extends to a covering map $S^1\to S^1$ of degree $e$; in particular $\deg(p_d|_X)=1$ if and only if $p_d|_X$ preserves the cyclic ordering of $X$. A cycle $C$ (viewed as a finite subset of $S^1$) is simple if $\deg(p_d|_C)=1$, i.e. the marking parametrizes it as a single winding; equivalently its points form an orbit whose cyclic order matches that of a rigid rotation. A finite set $X\subset S^1$ is renormalizable if it admits a nontrivial partition into disjoint unlinked subsets (subsets whose convex hulls in the disk are pairwise disjoint) that are permuted by the map; a collection of cycles $C_1,\dots,C_m$ of $p_d$ is binding if $\deg(p_d|_{\bigcup C_i})=d$ and $p_d|_{\bigcup C_i}$ is not renormalizable. Binding collections play the role, for circle dynamics, that filling multicurves — systems of closed curves that cut a surface into disks — play for hyperbolic surfaces. The space $\mathrm{QB}_d$ of quasi-Blaschke products consists of rational maps $f(z)=z\prod_{i=1}^{d-1}\frac{z-a_i}{1-b_i z}$ with $\prod_i|a_i|<1$ and $\prod_i|b_i|<1$, whose Julia set (the closed set of points at which the family of iterates of $f$ fails to be normal, i.e. the locus separating the regular basins from the chaotic dynamics) is a Jordan curve; such a map is obtained by gluing two Blaschke products $f_1,f_2\in\mathcal{B}_d$ along the circle, so expressions like $L(C_i,f_i)$ refer to the restriction on each side of the Jordan curve. $\mathcal{M}\mathrm{Rat}_d$ denotes the moduli space of rational maps of degree $d$ up to conjugation by Möbius transformations. The paper proves: (Theorem 1.1) any cycle with $L(C,f)<\log 2$ is simple, and all such short cycles share a rotation number; (Theorem 1.2) every $f\in\mathcal{B}_d$ has a simple cycle with $L(C,f)=O(d)$; (Theorem 1.3) for a binding collection $(C_i)$, the sublevel set $\{f\in\mathcal{B}_d:\sum_i L(C_i,f)\le M\}$ has compact closure in $\mathcal{M}\mathrm{Rat}_d$; (Theorem 1.4) the closure of the set of simple cycles has Hausdorff dimension zero. These are analogs of classical results for hyperbolic surfaces — surfaces carrying a complete metric of constant curvature $-1$, studied via their Teichmüller space, the space of marked conformal structures on the surface up to isotopy — due to Kerckhoff, Otal, and Thurston: closed geodesics shorter than $\log(3+2\sqrt{2})$ are simple and disjoint, every closed surface of genus $g$ has a simple closed geodesic of length $O(\log g)$, and a bound on the total length of a filling multicurve confines the surface to a compact part of Teichmüller space.

Problem Statement

As established in McMullen 2010 (Comment. Math. Helv. 85 (2010), 723-749), the following five questions are open. They are independent open problems: no one of them is implied by another, and settling any one of them is a separate result.

Q1. Let $C$ be a simple cycle. Is the function $L(C,f)$ free of critical points in $\mathcal{B}_d$?

Q2. Let $(C_i)$ be a binding collection of cycles. Does $\sum_i L(C_i,f)$ achieve its minimum at a unique point $f\in\mathcal{B}_d$?

Q3. Let $\mathrm{QB}_d$ denote the space of rational maps $f(z)=z\prod_{i=1}^{d-1}\frac{z-a_i}{1-b_i z}$ with $\prod_i|a_i|<1$ and $\prod_i|b_i|<1$, such that the Julia set of $f$ is a Jordan curve (equivalently, $f$ is obtained by gluing two Blaschke products $f_1,f_2\in\mathcal{B}_d$ along the circle). Does there exist $\delta_d>0$ such that, for all $f\in\mathrm{QB}_d$, all cycles $C$ with $L(C,f)<\delta_d$ are simple? (Here $L(C,f)$ denotes the cycle length computed for the rational map $f$ itself, on its Jordan Julia curve.)

Q4. Suppose the cycles $(C_1,C_2)$ are binding. Does the set of $f\in\mathrm{QB}_d$ with $L(C_1,f_1)+L(C_2,f_2)\le M$ have compact closure in the moduli space of all rational maps of degree $d$? (Here $f_1,f_2$ are the two Blaschke products glued along the circle to form $f$, and $L(C_i,f_i)$ is the cycle length computed on each side of the Jordan Julia curve.)

Q5. Is there a useful notion of intersection number for a pair of cycles?

All terms are as defined in the source paper: a cycle is a periodic orbit of the model map $p_d$ on $S^1$, its length is $L(C,f)=\log|(f^q)'(z)|$ where $q=|C|$ and $z$ is any point of the periodic orbit of $f$ marked by $C$ (i.e. any $z$ with $\phi_f(z)\in C$, for the canonical marking $\phi_f$ conjugating $f$ to $p_d$), a cycle is simple if $\deg(p_d|_C)=1$, where for a finite $p_d$-invariant set $X\subset S^1$ the degree $\deg(p_d|_X)$ is the least $e>0$ such that $p_d|_X$ extends to a covering map $S^1\to S^1$ of degree $e$ (so that $\deg(p_d|_X)=1$ if and only if $p_d|_X$ preserves the cyclic ordering of $X$), a finite subset of $S^1$ is renormalizable if it admits a nontrivial partition into disjoint unlinked subsets permuted by the map, and a collection of cycles $C_1,\dots,C_m$ of $p_d$ is binding if $\deg(p_d|_{\bigcup C_i})=d$ and the union is not renormalizable. The paper notes that the hyperbolic-surface analogs of these questions have positive answers, citing work of Kerckhoff, Otal, and Thurston: on a hyperbolic surface, geodesics shorter than $\log(3+2\sqrt{2})$ are simple and disjoint, and a bound on the lengths of a filling multicurve confines the surface to a compact part of Teichmüller space; for Q5 the surface counterpart is the geometric intersection number of simple closed curves.

References

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