Hausdorff dimension of Sierpiński carpet Julia sets: always strictly larger than one?

For a rational map whose Julia set is a Sierpiński carpet, the Hausdorff dimension is at least one, and Fu and Yang (2020) proved that carpet Julia sets realize every value in $(1,2)$, together with examples of dimension $2$ (with or without positive area) and abstract planar carpets — not coming from rational dynamics — of dimension exactly $1$. They asked the sharpness question at the lower endpoint: must every Sierpiński carpet Julia set of a rational map have Hausdorff dimension strictly larger than one? The strict bound is proven when the map has an attracting or parabolic basin, or is renormalizable around a sufficiently high-type Siegel disk; the remaining case, maps whose periodic Fatou components are all Siegel disks (where the post-critical set is uncontrolled), is open, as is the existence of any carpet Julia set of dimension exactly $1$.

Background

Holomorphic dynamics studies the iteration of a rational map $f$ (a quotient of two polynomials) on the Riemann sphere. The sphere splits into the Fatou set, where the family of iterates $\{f^n\}$ is normal and the behavior is regular, and its complement, the Julia set $J(f)$, the locus of chaotic dynamics. A Sierpiński carpet is a planar set homeomorphic to the standard middle-thirds carpet; by a classical criterion of Whyburn, a Julia set is a Sierpiński carpet precisely when it is compact, connected, locally connected, has empty interior, and all Fatou components are Jordan domains with pairwise disjoint boundaries. Carpet Julia sets occur in explicitly describable families, notably the McMullen maps $f_\lambda(z)=z^n+\lambda/z^n$, $n\ge 3$.

Hausdorff dimension $\dim_H$ measures the scaling of covers by small sets and refines topological dimension. Every continuum (compact connected set) with at least two points satisfies $\dim_H\ge 1$, so for a carpet Julia set the only alternative to $\dim_H J(f)>1$ is $\dim_H J(f)=1$ exactly.

The neighboring results frame the question. Buff and Chéritat, and later Avila and Lyubich, constructed quadratic Julia sets of positive area (hence of dimension $2$). Haïssinsky and Pilgrim produced carpet Julia sets whose conformal dimension — the infimum of Hausdorff dimension over all quasisymmetric images — is arbitrarily close to $2$. Barański and Wardal showed that for McMullen maps with Sierpiński carpet Julia sets, $\dim_H J=1+O(1/\ln n)$, so these dimensions approach $1$ from above. Fu and Yang then proved: carpet Julia sets of positive area exist, containing a Cremer fixed point, a Siegel disk, or infinite renormalization (Theorem A); zero-area carpet Julia sets of dimension $2$ exist (Theorem B); for every $s\in(1,2)$ there is a carpet Julia set with $\dim_H=s$ exactly (Theorem C); and there exist abstract planar Sierpiński carpets, not arising from any rational dynamics, with $\dim_H=1$ (Theorem D, via a construction of H. Ruan) — but these are far from self-similar.

A few dynamical notions are needed for the partial results. A Siegel disk is a Fatou component on which $f$ is conformally conjugate to an irrational rotation; its boundary lies in the Julia set and can be a smooth curve (dimension $1$, by results of Avila, Buff and Chéritat) or can have dimension strictly between $1$ and $2$. A Cremer point is an irrationally indifferent periodic point at which the map is not locally linearizable. A map $f$ is renormalizable if some iterate $f^p$ restricted to a domain pair $(U,V)$ is polynomial-like of degree $2$ with connected filled Julia set; $f$ is infinitely renormalizable when this happens for unbounded periods. The post-critical set is the closure of the forward orbits of the critical points. An irrational rotation number is of high type $N$ if all continued-fraction partial quotients are at least $N$.

Fu and Yang's Theorem 5.1 proves $\dim_H J(f)>1$ whenever $f$ has an attracting basin or a parabolic basin (via results of Przytycki and Urbański giving dimension $>1$ for such basin boundaries, which are subsets of the Julia set), or when $f$ is renormalizable and the small filled Julia set contains a sufficiently high-type quadratic Siegel disk — in that case the post-critical set has dimension $2$, forcing $\dim_H J(f)=2$. The situation not covered is a carpet map whose periodic Fatou components are all Siegel disks: there the post-critical set cannot currently be controlled, the map may be non-renormalizable, and Siegel disk boundaries may themselves have dimension $1$, so no single Fatou-component boundary supplies the gap. The authors nevertheless expect an affirmative answer, on the heuristic that Julia sets possess a self-similarity (even around Siegel disk boundaries) that the dimension-$1$ abstract carpets of Theorem D lack.

Problem Statement

Let $f$ be a rational map on the Riemann sphere whose Julia set $J(f)$ is a Sierpiński carpet. Since $J(f)$ is a continuum, $\dim_H J(f)\ge 1$ holds automatically. Decide whether $\dim_H J(f)>1$ for every such rational map; equivalently, either prove that no rational map has a Sierpiński carpet Julia set of Hausdorff dimension exactly $1$, or construct a rational map whose Julia set is a Sierpiński carpet with $\dim_H J(f)=1$. The case left open by the known partial results (Fu–Yang Theorem 5.1) is a carpet map with no attracting or parabolic basin and no renormalization around a controlled high-type Siegel disk — in particular, a map whose periodic Fatou components are all Siegel disks.

References

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