Lebesgue measure of the set of infinitely renormalizable parameters in the Multibrot set $\mathcal{M}_d$

For the unicritical family $f_c(z) = z^d + c$ ($d \geq 2$), Avila, Lyubich and Shen proved that Lebesgue-almost every complex parameter $c$ is either hyperbolic, meaning the critical orbit is attracted to a periodic cycle, or infinitely renormalizable, which reduces the measure-theoretic Fatou conjecture for this family to the size of the exceptional set. The problem is to determine the two-dimensional Lebesgue measure (area) of the set $\mathcal{I}_d$ of infinitely renormalizable parameters in the degree-$d$ connectedness locus (Multibrot set) $\mathcal{M}_d$, where $\mathcal{M}_2$ is the Mandelbrot set: prove that $\operatorname{area}(\mathcal{I}_d) = 0$, or prove that it is positive, for a degree $d \geq 2$. By the dichotomy theorem, zero area is equivalent to the statement that almost every complex parameter is hyperbolic. The question is open for every $d \geq 2$. The real one-dimensional slice is settled for quadratics (Lyubich, 2002) and treated as settled in higher even degree through the Avila-Lyubich renormalization horseshoe and Clark's hybrid-class lamination. The complex-parameter case remains open.

Background

Fix an integer $d \geq 2$ and consider the unicritical family of polynomials $f_c(z) = z^d + c$, parametrized by $c \in \mathbb{C}$; each map has a single critical point, $z = 0$ (a critical point is a point where the derivative vanishes). The Julia set of $f_c$ is the boundary between the set of starting points whose orbits under iteration stay bounded and the set of those that escape to infinity, and the connectedness locus $\mathcal{M}_d = \{ c \in \mathbb{C} : \text{the Julia set of } f_c \text{ is connected} \}$ is the degree-$d$ Multibrot set; $\mathcal{M}_2$ is the Mandelbrot set. A map $f_c$ is called hyperbolic if its critical orbit converges to an attracting periodic orbit; hyperbolic maps are dynamically the simplest, and their parameters form an open set. The classical hyperbolicity conjecture of Fatou asserts that hyperbolic maps are dense in the family, and its measure-theoretic form asserts that Lebesgue-almost every $c \in \mathbb{C}$ is hyperbolic.

Renormalization. A polynomial-like map, in the standard sense of Douady and Hubbard, is a proper holomorphic map $g : U \to V$ between topological disks with $U \Subset V$; by their straightening theorem such a map is quasiconformally (i.e., by a homeomorphism with bounded infinitesimal distortion) conjugate to a polynomial. A unicritical map $f_c$ is renormalizable if some iterate $f_c^p$ restricted to a disk pair around the critical point is a polynomial-like map with connected Julia set, and infinitely renormalizable if such renormalizations exist to arbitrarily deep levels (with unbounded periods $p$). Infinitely renormalizable maps are never hyperbolic (the critical orbit is trapped in a nested sequence of small Julia sets). A priori bounds - the property that the small Julia sets remain comparable in size at all renormalization depths - can fail, and the parameters without a priori bounds are the least understood part of the set $\mathcal{I}_d$ of infinitely renormalizable parameters.

The source paper. Avila, Lyubich and Shen (J. Eur. Math. Soc. 13 (2011) 27-56) studied typical dynamics in the unicritical family. For real parameters and even $d$ (so that $f_c$ restricted to an invariant interval is a unimodal map, that is, a smooth interval map with a single interior turning point) they proved the trichotomy (their Theorem 1.1) that for almost every real parameter the map is either regular, meaning the critical orbit is attracted to a periodic sink, or Collet-Eckmann, meaning the derivative along the critical orbit grows exponentially fast, which entails stochastic dynamics with an absolutely continuous invariant probability measure, or infinitely renormalizable; their Remark 1.3 states that the set of infinitely renormalizable real parameters has zero Lebesgue measure - completing the Regular or Stochastic dichotomy - and defers the proof of this zero-measure statement in higher even degree to forthcoming notes. For complex parameters they proved (their Theorem 1.3) that for Lebesgue-a.e. $c \in \mathbb{C}$ the map $f_c$ is either hyperbolic or infinitely renormalizable, and (their Theorem 1.4, with footnote 3 and the sketch in Remark 6.1) that a parameter whose map is non-renormalizable with all fixed points repelling is not a Lebesgue density point of $\mathcal{M}_d$ (a density point being a point where the set occupies asymptotically full relative area in arbitrarily small disks), the finitely renormalizable extension being only indicated.

Prior and adjacent results. For the quadratic family, Lyubich (Ann. of Math. 156 (2002) 1-78) proved that the set of real infinitely renormalizable parameters has zero one-dimensional Lebesgue measure, completing the Regular or Stochastic dichotomy for real quadratics; Avila, Lyubich and de Melo (Invent. Math. 154 (2003) 451-550) had earlier extended the dichotomy to quasiquadratic analytic families. The higher-degree real case rests on the Avila-Lyubich renormalization horseshoe (Publ. Math. IHÉS 114 (2011) 171-223): a precompact invariant horseshoe for the renormalization operator on unicritical maps of any degree, exponentially contracting along hybrid classes (quasiconformal equivalence classes with $\bar\partial$-derivative vanishing a.e. on the Julia set). The transverse unstable direction needed to convert this into the parameter-measure statement was deferred by that paper to a further forthcoming paper; Clark (Ergodic Theory Dynam. Systems 34 (2014) 1538-1566) constructed a lamination (a foliation-like partition) of the space of unimodal maps of fixed critical degree by hybrid classes and used it to transfer the Regular or Stochastic dichotomy to arbitrary non-trivial real analytic families of such maps. In the complex parameter plane, Shishikura (Ann. of Math. 147 (1998) 225-267) proved that the boundary of the Mandelbrot set has Hausdorff dimension 2, while Buff and Chéritat (Ann. of Math. 176 (2012) 673-746) constructed quadratic Julia sets of positive area in the dynamical plane - a warning that measure statements in this circle of problems can fail in the absence of a specific mechanism.

Problem Statement

Determine the two-dimensional Lebesgue measure (area) of the set of infinitely renormalizable parameters in the unicritical family.

For an integer $d \geq 2$, let $f_c(z) = z^d + c$ and let $\mathcal{M}_d$ be the degree-$d$ connectedness locus (the Multibrot set; $\mathcal{M}_2$ is the Mandelbrot set). Let $\mathcal{I}_d \subset \mathcal{M}_d$ be the set of parameters $c$ such that $f_c$ is infinitely renormalizable, that is, there exist pairs of topological disks $(U_n, V_n)$ with $U_n \Subset V_n$ around the critical point $z = 0$ and integers $1 < p_1 < p_2 < \cdots$ with $p_n \to \infty$ such that $f_c^{p_n} : U_n \to V_n$ is a polynomial-like map (a proper holomorphic map between topological disks with $U_n \Subset V_n$) with connected Julia set for every $n$.

The problem: prove that $\operatorname{area}(\mathcal{I}_d) = 0$, or prove that $\operatorname{area}(\mathcal{I}_d) > 0$, for a degree $d \geq 2$; the strongest form covers all $d \geq 2$, and the quadratic case $d = 2$ alone is a complete answer for that degree. By Theorem 1.3 of Avila-Lyubich-Shen (2011) - for Lebesgue-almost every $c \in \mathbb{C}$ the map $f_c$ is either hyperbolic (the critical orbit converges to an attracting periodic orbit) or infinitely renormalizable - together with the standard fact that an infinitely renormalizable unicritical map is not hyperbolic, the statement $\operatorname{area}(\mathcal{I}_d) = 0$ is equivalent to the statement that Lebesgue-almost every complex parameter $c$ of the family $z \mapsto z^d + c$ is hyperbolic, the measure-theoretic form of Fatou's hyperbolicity conjecture for this family. A proof of positive area for any $d$ would refute that conjecture for that degree.

Scope requirements. The answer must cover the entire set $\mathcal{I}_d$ for the stated degree: infinitely renormalizable parameters of satellite combinatorics (the small Julia set of the renormalization touches the original Julia set at a repelling periodic point) and of primitive combinatorics (the small Julia set is separated from the original one), with or without a priori bounds (uniform size control of the small Julia sets across renormalization depths), and with arbitrarily deep decorations (nested small copies of $\mathcal{M}_d$ accumulating at the parameter). Partial results restricted to subclasses - for example zero area of the parameters with a priori bounds, zero area of bounded-combinatorics parameters, zero one-dimensional measure of the real slice $\mathcal{I}_d \cap \mathbb{R}$, Hausdorff-dimension bounds, or porosity and density-point statements - are welcome intermediate progress but do not by themselves settle this question.

References

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