Bounded escape as a component property: bounded multipliers and the uniform extremal-annulus modulus in nested hyperbolic components of rational maps (Luo, Questions 11.14–11.15)

For a hyperbolic component $\mathcal{H}$ of the moduli space of degree-$d$ rational maps that admits bounded escape — equivalently, by a theorem of Luo, one whose Julia sets are nested (McMullen-type Cantor sets of circles) — the source paper poses two open questions and asserts their equivalence: (Question 11.14) must every degenerating sequence in $\mathcal{H}$ have bounded multipliers, that is, is the marked length spectrum uniformly bounded on $\mathcal{H}$, so that bounded escape is a property of the whole component rather than of one particular sequence; and (Question 11.15) does the modulus $M(f)$ of the annulus bounded by the two extremal Julia components admit a uniform positive lower bound over $\mathcal{H}$.

Background

A rational map $f:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ of degree $d\ge 2$ is called hyperbolic if every critical point of $f$ converges to an attracting periodic cycle under iteration. Hyperbolic maps form an open subset of the moduli space $\mathcal{M}_d$ of rational maps modulo conjugacy by Möbius transformations, and a connected component $\mathcal{H}$ of this set is called a hyperbolic component. Each hyperbolic component carries a topological model $\sigma:J\to J$ of the dynamics on the Julia set $J(f)$ (the closure of the repelling periodic points, on which the chaotic dynamics concentrates): for every $[f]\in\mathcal{H}$ there exists a marking, a homeomorphism $\varphi:J\to J(f)$ conjugating $\sigma$ to $f$. Given a marking, the marked length spectrum assigns to each periodic cycle $C$ of the model the quantity $L(C,[f])=\log|(f^{q})'(z)|$, where $q=|C|$ and $z\in\varphi(C)$ is the corresponding periodic point; this is the logarithm of the multiplier (the derivative of the iterate) of the marked repelling cycle. It plays the role that the marked length spectrum of closed geodesics plays for a hyperbolic 3-manifold.

A sequence $[f_n]\in\mathcal{H}$ is degenerating if it eventually leaves every compact subset of $\mathcal{M}_d$. In the classical analogy with degenerating Kleinian groups, degeneration forces the length spectrum to explode. For rational maps this fails: McMullen's family $f_\lambda(z)=z^{2}+\lambda/z^{3}$ lies in one hyperbolic component for small $\lambda$, degenerates in $\mathcal{M}_5$ as $\lambda\to 0$ (the limit $z^{2}$ has lower degree, so no degree-$5$ conjugacy class is approached), yet all multipliers stay bounded; the Julia set is a Cantor set of Jordan curves. Luo defines a hyperbolic component $\mathcal{H}$ to admit bounded escape if there is a degenerating sequence $[f_n]\in\mathcal{H}$ with $L(C,[f_n])$ bounded for every periodic cycle $C$ of the model, equivalently (his condition (2$'$)) with the multipliers of periodic points of any fixed period staying bounded. He calls $J(f)$ nested if it has more than one component and there are two points $p_1,p_2$ separated by every component of $J(f)$, and proves (Theorem 1.5) that a hyperbolic component admits bounded escape if and only if it has nested Julia sets; bounded-escape components are thus exactly the Cantor-circle-type components, classified by combinatorial tree-mapping-scheme data.

For a nested Julia set, a Julia component $K$ is extremal if the remainder $J\setminus K$ of the Julia set is contained in a single component of the complement $\hat{\mathbb{C}}\setminus K$; there are exactly two extremal components, the innermost and outermost curves of the nested configuration, and $M(f)$ denotes the modulus of the annulus they bound — the modulus of an annulus being the conformal invariant equal to $\log(r_2/r_1)/(2\pi)$ for a round annulus $\{r_1<|z|<r_2\}$, carried by every doubly connected domain via uniformization. Luo's quantitative Theorem 11.13 shows that in a hyperbolic component without nested Julia sets, every degenerating sequence has some marked length growing to infinity at a rate comparable to his degeneration scale $r([f_n])$ (defined through barycentric extensions of rational maps to hyperbolic $3$-space, the construction underlying his limiting dynamics on $\mathbb{R}$-trees): off the nested locus, the multiplier spectrum explodes. What the theory does not decide is the behavior inside the nested locus itself. Luo closes the paper with the questions stated below, together with the observations that make them equivalent: if the moduli $M(f_n)$ are bounded from below along a degenerating sequence, then an argument as in his Theorem 11.1 keeps the lengths of every periodic cycle bounded; conversely, a degenerating sequence with bounded length spectrum forces $M(f_n)\to\infty$ (from the proof of his Theorem 11.11). Hence an unbounded length spectrum inside $\mathcal{H}$ could only arise through collapse of the extremal annulus, $M(f_n)\to 0$.

Problem Statement

Let $\mathcal{H}\subset\mathcal{M}_d$ be a hyperbolic component of the moduli space of rational maps of degree $d\ge 2$ that admits bounded escape; equivalently, by Luo's Theorem 1.5, a hyperbolic component with nested Julia sets.

(1) (Question 11.14) Must every degenerating sequence $[f_n]\in\mathcal{H}$ have bounded multipliers — that is, do the multipliers of the periodic points of $f_n$ of any fixed period stay bounded along every degenerating sequence, so that the marked length spectrum is uniformly bounded over $\mathcal{H}$ and bounded escape is a property of the component $\mathcal{H}$ rather than of one particular sequence in it?

(2) (Question 11.15) Equivalently, as asserted by Luo: does there exist a constant $\varepsilon>0$, depending on $\mathcal{H}$, such that $M(f)\ge\varepsilon$ for every $[f]\in\mathcal{H}$, where $M(f)$ is the modulus of the annulus bounded by the two extremal Julia components of $J(f)$?

The problem is to resolve either formulation in full generality: prove the uniform multiplier bound (equivalently the uniform positive lower bound on $M(f)$) for every hyperbolic component with nested Julia sets, or else construct, in some hyperbolic component with nested Julia sets, a degenerating sequence along which some periodic multiplier tends to infinity (equivalently, a sequence $[f_n]\in\mathcal{H}$ with $M(f_n)\to 0$).

References

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