Sierpiński Carpet Julia Sets

• Sierpiński carpet Julia sets are fractal sets defined by rational map dynamics that are compact, locally connected, and feature disjoint Jordan curve boundaries for their Fatou components.
• They emerge from specific dynamical conditions in postcritically finite or hyperbolic maps, such as McMullen and quadratic rational maps, through controlled combinatorial and analytic criteria.
• Quasisymmetric rigidity results ensure that self-homeomorphisms of these sets reduce to Möbius transformations, highlighting their geometric invariance and complex measure-theoretic behavior.

A Sierpiński carpet Julia set is a dynamically generated fractal arising as the Julia set $J(f)$ of a rational map $f:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$, homeomorphic to the standard Sierpiński carpet in the sense of Whyburn, i.e., a compact, connected, locally connected subset of the sphere with empty interior whose complementary domains are bounded by pairwise disjoint Jordan curves shrinking to points. Such Julia sets typically occur for postcritically finite (PCF) or hyperbolic rational maps with highly controlled combinatorics and critical orbit arrangements.

1. Topological and Dynamical Characterization

The topological definition, via Whyburn, specifies that $J(f)$ is a Sierpiński carpet if it is compact, connected, locally connected, nowhere dense, and its complement is a countable union of open Jordan domains whose boundaries are simple closed curves, pairwise disjoint, with diameters tending to zero (Gao et al., 2015). The Fatou set $\mathcal{F}(f)$ consists of these domains, each corresponding to a Fatou component, typically a disk whose closure remains disjoint from all other Fatou components.

For PCF rational maps, the equivalence between $J(f)$ being a Sierpiński carpet and $f$ being Thurston equivalent to an expanding Thurston map is established. These expanding Thurston maps are characterized by the shrinking of "tiles" (preimages of Jordan curves through the postcritical set) under iteration, ensuring uniform expansion on the sphere relative to a visual metric (Gao et al., 2015).

In the context of quadratic rational maps, Sierpiński carpet Julia sets arise precisely in certain hyperbolic parameter regimes (types C and D in the Milnor–Lei–Rees classification) where the two critical points of $f$ have orbits that avoid sharing boundaries or create only isolated periodic boundary contacts. For hyperbolic maps with no attracting fixed point, the closure of every Fatou component is a Jordan disk, and any two distinct Fatou components have disjoint closures—essential for the Sierpiński carpet property (Devaney et al., 2011).

2. Criteria for Existence and Families Realizing Carpet Julia Sets

Fei Yang's criterion provides a sharp, verifiable set of dynamical and local degree conditions sufficient to ensure that a rational map $f$ of degree at least three has a carpet Julia set (Yang, 2017). The criterion requires:

• $J(f)$ connected,
• The existence of two distinct Fatou components $U$ and $V$ with $f(U)=U$ and $f^{-1}(U)=U\cup V$,
• Every component $W$ of $f^{-1}(V)$ mapping with higher degree than $U\to U$,
• Every critical orbit eventually entering $U$.

This condition efficiently covers canonical hyperbolic families, McMullen maps ($f_\lambda(z) = z^m + \lambda/z^\ell$), the Morosawa-Pilgrim family, and others. It also applies to special examples with parabolic basins or super-attracting cycles, and even some with infinitely renormalizable or non-hyperbolic behavior, provided local connectivity and disjointness of Fatou boundaries are maintained (Yang, 2017).

For the McMullen family, detailed in (Qiu et al., 2013), the escape trichotomy classifies the topology of $J(f_\lambda)$ based on the behavior of its "free" critical points:

• When all free critical points escape to $\infty$, $J_\lambda$ is either a Cantor set, a Cantor set of circles, or a Sierpiński carpet.
• When they do not escape, sufficient conditions (primitive hyperbolic components, absence of satellites) guarantee $J_\lambda$ is a quasisymmetrically round carpet.

3. Quasisymmetric and Geometric Rigidity

Rigidity phenomena in the context of Sierpiński carpet Julia sets are profound. Bonk–Lyubich–Merenkov established that for PCF rational maps with Sierpiński carpet Julia sets, every quasisymmetric homeomorphism between such Julia sets is the restriction of a Möbius transformation; thus, the quasisymmetry group is finite (Bonk et al., 2014). The geometric mechanism hinges on the properties of the carpet:

• Peripheral circles (boundaries of the Fatou components) are uniform quasicircles, uniformly relatively separated, and distributed on all locations and scales.
• The locally porous nature of the carpet ensures that peripheral circles occur at all scales near every point (Qiu et al., 2014).
• Quasisymmetries preserve this geometry, limiting the possible deformations to conformal automorphisms.

This rigidity contrasts strongly with the flexibility seen for Julia sets of other topological types (e.g., dendrites or gaskets), where quasisymmetry groups can be infinite or include Thompson-type groups (Merenkov et al., 26 Jan 2026). In fact, every quasiregular self-map of the sphere carrying the Julia set onto itself must be rational and satisfy a commuting relation with $f$, cementing an absolute rigidity unique to the Sierpiński carpet setting.

4. Explicit Constructions and Parametric Aspects

Sierpiński carpet Julia sets are realized in several classic families, with explicit parameter regimes and normal forms. For the McMullen family, parameters yielding Sierpiński carpet topology are well-understood both in escaping and non-escaping loci, with the Mandelbrot copy in parameter space controlling more pathological cases (infinitely renormalizable, Siegel, or Cremer interiors) (Qiu et al., 2013, Fu et al., 2018).

Quadratic rational maps realize Sierpiński carpets when, in suitable one-parameter slices (e.g., Per$_n$(0)), their critical orbits avoid any persistent low-period boundary collisions among Fatou components (Devaney et al., 2011). Explicit normal forms (e.g., $f_a(z) = (z-1)((z-a)/(2-a))/z^2$) and parameter slices allow identification of the relevant Sierpiński regions in moduli space.

In addition, the construction techniques extend via polynomial-like renormalization to embed quadratic Julia sets with various properties (Siegel disks, Cremer points, Hausdorff dimension two) into the global Julia set of a higher-degree rational map, thereby realizing "wild" carpet Julia set properties, including positive area and full Hausdorff dimension (Fu et al., 2018).

5. Combinatorial and Geometric Tools

Proof strategies exploit several deep tools:

• Bonk's round uniformization: If the peripheral circles are uniform quasicircles and relatively separated, the carpet is quasisymmetrically equivalent to the round Sierpiński carpet.
• Markov partitions via invariant Jordan curves: Every PCF rational map with a carpet Julia set admits an invariant Jordan curve containing the postcritical set, leading to combinatorial Markov partitions akin to those for expanding Thurston maps (Gao et al., 2015).
• Conformal dynamical estimates: Techniques such as the "conformal elevator," distortion control, and modulus estimates underlie the verification that Fatou boundaries are quasicircles and that their separation is uniform, ensuring the roundness criteria.
• Schottky set rigidity: Within the class of relative Schottky sets (unions of round disks), any locally quasisymmetric self-homeomorphism is conformal (Bonk et al., 2014), supporting the uniqueness and stability of carpet Julia sets under geometric perturbations.

6. Measure and Dimension Theory

Fu–Yang demonstrated the possible fractal-geometric diversity of carpet Julia sets: there exist rational maps whose Julia sets are Sierpiński carpets of either positive area or zero area but full Hausdorff dimension two. For any $s\in(1,2)$ there is a rational map with Sierpiński carpet Julia set of Hausdorff dimension $s$, realized through analytic variation in hyperbolic parameters of McMullen maps, together with appropriate polynomial-like insertions (Fu et al., 2018).

Notably, this breaks the previous dichotomy where all known carpet Julia sets were of measure zero (even if $\dim_H=2$), and expands the range of pathological behaviors possible for Julia sets within the carpet topology.

7. Connections, Open Directions, and Classification

Sierpiński carpet Julia sets bridge several advanced domains:

• Expanding Thurston maps: The equivalence between Sierpiński carpet Julia sets and expanding Thurston maps provides a transfer of techniques and combinatorial frameworks between complex dynamics and topological dynamics (Gao et al., 2015).
• Combinatorial classification: The dynamical types of critical orbit organization in the moduli of rational maps sharply classify the possible existence and geometry of carpet Julia sets, particularly in quadratic and McMullen-type families (Devaney et al., 2011, Qiu et al., 2013).
• Automorphism and rigidity theory: The precise finiteness and rigidity results for the symmetry groups of carpet Julia sets distinguish them from limit sets of Kleinian groups or more flexible Julia set types (Bonk et al., 2014, Merenkov et al., 26 Jan 2026).
• Measure-theoretic and parametric flexibility: The explicit realization of full dimension and measure spectrum within carpets sets a template for subsequent explorations in rational dynamics (Fu et al., 2018).

Open questions include the classification and uniqueness of invariant Jordan curves (Markov partitions), the structure of parameter space loci supporting carpet topology, mating and surgery operations on carpet Julia sets, and the ultimate boundary between topological and geometric rigidity in more general dynamical settings (Gao et al., 2015, Qiu et al., 2014).