Canonical Crochet–Sierpiński Decomposition

• The Canonical Crochet–Sierpiński Decomposition is a framework that splits Böttcher expanding maps into two distinct regimes based on the topology of Fatou component closures.
• It employs a unique invariant multicurve to delineate crochet pieces—where components collapse to points—from Sierpiński pieces that preserve sphere-like structures.
• The approach yields a maximal expanding quotient and an effective, recursive Crochet Algorithm, offering actionable insights into rational map dynamics.

Canonical Crochet–Sierpiński Decomposition is a canonical decomposition theory for postcritically finite rational maps with non-empty Fatou set, developed in the broader setting of Böttcher expanding maps. It separates the dynamics into two topological regimes—Sierpiński behavior, where Fatou component closures are Jordan disks with pairwise disjoint closures, and crochet behavior, where Fatou components are linked by countable chains of touching boundaries—and encodes that separation by a unique invariant multicurve and a canonical quotient cactoid. In this formulation, crochet pieces collapse to points, Sierpiński pieces survive as small spheres, and the induced quotient map is the maximal totally expanding quotient (Dudko et al., 2022).

1. Scope and formal setting

The decomposition is formulated for Böttcher expanding maps $f\colon (S^2,A)$, a class that contains postcritically finite rational maps with non-empty Fatou set. In this setting, the Julia set $\mathcal J(f)$ is the closure of the repelling periodic points, the Fatou set is $\mathcal F(f)=S^2\setminus \mathcal J(f)$, and each Fatou component is an open topological disk, eventually periodic, and carries Böttcher coordinates (Dudko et al., 2022).

Its organizing principle is the topology of Fatou component closures. If closures of Fatou components are pairwise disjoint, the resulting behavior is Sierpiński. If closures touch along boundary points and form chains, the resulting behavior is crochet. The decomposition is therefore controlled by how closures of Fatou components intersect, rather than by an a priori combinatorial subdivision (Dudko et al., 2022).

A central object is the Fatou-collapse equivalence relation

$\sim_{\mathcal F(f)},$

defined as the smallest closed equivalence relation generated by identifying all points in every Fatou component. The quotient

$S^2/\sim_{\mathcal F(f)}$

is a cactoid: a locally connected continuum made of countably many spheres and segments, any two meeting in at most one point. The induced map

$$\bar f\colon S^2/\sim_{\mathcal F(f)}\to S^2/\sim_{\mathcal F(f)}$$

is totally topologically expanding (Dudko et al., 2022).

This already indicates the sense in which the decomposition is canonical. The equivalence relation is intrinsic, the quotient is dynamically natural, and the residual expanding structure is read off from Fatou topology itself rather than from an arbitrary choice of graph or puzzle.

2. The Sierpiński–crochet dichotomy

A Böttcher expanding map is a Sierpiński map if its Julia set is homeomorphic to the standard Sierpiński carpet; equivalently, $\mathcal F(f)\neq\emptyset$, every Fatou component closure is a Jordan domain, and closures of distinct Fatou components are pairwise disjoint. The paper also gives equivalent quotient and graph formulations: $S^2/\sim_{\mathcal F(f)}$ is a sphere and $\sim_{\mathcal F(f)}$ is trivial on $A$, and every connected periodic zero-entropy graph is homotopically trivial rel.\ $\mathcal J(f)$0 (Dudko et al., 2022).

A Böttcher expanding map is a crochet map if there exists a connected forward-invariant zero-entropy graph $\mathcal J(f)$1 containing $\mathcal J(f)$2. The paper gives two further equivalent characterizations: $\mathcal J(f)$3 is a singleton, and every two points in $\mathcal J(f)$4 can be connected by a path $\mathcal J(f)$5 with $\mathcal J(f)$6 countable (Dudko et al., 2022).

These two regimes are topological opposites. In the Sierpiński regime, collapsing each Fatou component separately preserves a sphere. In the crochet regime, collapsing the Fatou components forces the entire configuration to a point. The decomposition theorem asserts that a general map in the class splits canonically into pieces of these two kinds rather than belonging globally to only one regime (Dudko et al., 2022).

A common misconception is that a sphere quotient automatically means the whole map is Sierpiński. The examples discussed in the paper exclude that simplification: a mixed map can have one crochet piece and one Sierpiński piece while the global quotient is still a sphere, because the quotient being a sphere does not by itself imply that the equivalence relation is trivial on the postcritical set (Dudko et al., 2022).

3. Canonical invariant multicurve and the decomposition theorem

The decomposition is realized along a canonical invariant multicurve

$\mathcal J(f)$7

called the crochet multicurve. Cutting the marked sphere along $\mathcal J(f)$8 produces finitely many small spheres, and periodic cycles of small spheres determine small maps. The main theorem states that there is a unique canonical invariant multicurve whose small maps are Sierpiński or crochet, such that under the quotient map

$\mathcal J(f)$9

the small Julia sets of Sierpiński maps project onto spheres, the small Julia sets of crochet maps project to points, and different crochet Julia sets project to different points (Dudko et al., 2022).

The decomposition is simultaneously three things. It is a topological decomposition of the marked sphere by an invariant multicurve, a dynamical decomposition into first-return maps on periodic small spheres, and a quotient decomposition visible in the cactoid $\mathcal F(f)=S^2\setminus \mathcal J(f)$0 (Dudko et al., 2022).

The paper gives an alternative structural characterization in terms of the directed graph $\mathcal F(f)=S^2\setminus \mathcal J(f)$1 on an invariant multicurve. Strongly connected components are either unicycles or bicycles. A bicycle is a maximally strongly connected periodic family of curves that replicates under pullback: for some $\mathcal F(f)=S^2\setminus \mathcal J(f)$2, every curve is homotopic to at least two components of $\mathcal F(f)=S^2\setminus \mathcal J(f)$3. Maximal Sierpiński small maps are well-defined, and if two bicycles have positive geometric intersection number, then these bicycles lie within a small Sierpiński map. The crochet multicurve is generated by the boundaries of maximal Sierpiński small maps and the remaining bicycles (Dudko et al., 2022).

The decomposition also has a Sierpiński-free criterion. A map is Sierpiński-free if no invariant multicurve decomposition produces a Sierpiński small map, and this is equivalent to saying that none of the small maps in the decomposition along $\mathcal F(f)=S^2\setminus \mathcal J(f)$4 is Sierpiński and that $\mathcal F(f)=S^2\setminus \mathcal J(f)$5 is a dendrite (Dudko et al., 2022).

4. Construction from clusters of touching Fatou components

The construction begins with maximal clusters of touching Fatou components. At level $\mathcal F(f)=S^2\setminus \mathcal J(f)$6, pre-clusters are Fatou components and clusters are their closures. Inductively, level $\mathcal F(f)=S^2\setminus \mathcal J(f)$7 clusters are obtained by grouping level $\mathcal F(f)=S^2\setminus \mathcal J(f)$8 clusters that can be connected by a finite chain of intersections. The process stabilizes: there exists $\mathcal F(f)=S^2\setminus \mathcal J(f)$9 such that

$\sim_{\mathcal F(f)},$0

These are the maximal clusters of touching Fatou components (Dudko et al., 2022).

If $\sim_{\mathcal F(f)},$1 is the union of maximal clusters, the multicurve

$\sim_{\mathcal F(f)},$2

is formed by the boundary curves of complementary components of $\sim_{\mathcal F(f)},$3, and the paper proves that this multicurve is invariant. Periodic nontrivial clusters lie inside periodic small spheres of the decomposition along $\sim_{\mathcal F(f)},$4, and the corresponding first-return maps are crochet. Repeating this extraction inside pieces not yet classified yields a recursive decomposition (Dudko et al., 2022).

After recursively extracting all maximal crochet clusters one obtains a pre-crochet multicurve $\sim_{\mathcal F(f)},$5. At that stage all small maps are already either crochet or Sierpiński, but some primitive unicycles may still separate crochet pieces from crochet pieces. These are removed by a reduction step. The final crochet multicurve $\sim_{\mathcal F(f)},$6 is obtained by iteratively eliminating all primitive crochet unicycles; equivalently, $\sim_{\mathcal F(f)},$7 is generated by the boundaries of Sierpiński small spheres and bicycles (Dudko et al., 2022).

The paper states the Crochet Algorithm explicitly:

1. Compute maximal clusters of touching Fatou components and their boundary multicurve.
2. Decompose the map with respect to that boundary multicurve.
3. Repeat Steps 1–2 for each small map until all small maps are crochet or Sierpiński.
4. Glue small crochet maps that correspond to the same point in $\sim_{\mathcal F(f)},$8.

The algorithm is symbolic when the input is a sphere biset of

$\sim_{\mathcal F(f)},$9

and the paper emphasizes effective computability. Termination follows because the recursive steps are detected on the finite marked set and only finitely many separating relations are relevant (Dudko et al., 2022).

5. Maximal expanding quotient and cactoid dynamics

The decomposition is inseparable from the maximal expanding quotient theorem. For a Böttcher expanding map $S^2/\sim_{\mathcal F(f)}$0 with $S^2/\sim_{\mathcal F(f)}$1, the quotient map

$S^2/\sim_{\mathcal F(f)}$2

is the maximal totally expanding quotient. Any semiconjugacy from $S^2/\sim_{\mathcal F(f)}$3 to a totally topologically expanding map factors through $S^2/\sim_{\mathcal F(f)}$4, and

$S^2/\sim_{\mathcal F(f)}$5

Equivalently,

$S^2/\sim_{\mathcal F(f)}$6

Thus collapsing Fatou components is the largest quotient compatible with total topological expansion (Dudko et al., 2022).

The cactoid viewpoint makes the geometry of the decomposition explicit. Given an invariant multicurve $S^2/\sim_{\mathcal F(f)}$7, one decomposes

$S^2/\sim_{\mathcal F(f)}$8

where $S^2/\sim_{\mathcal F(f)}$9 consists of curves collapsed to points, $$\bar f\colon S^2/\sim_{\mathcal F(f)}\to S^2/\sim_{\mathcal F(f)}$$0 consists of curves thickened and collapsed to segments, $$\bar f\colon S^2/\sim_{\mathcal F(f)}\to S^2/\sim_{\mathcal F(f)}$$1 indexes crochet small spheres collapsed to points, and $$\bar f\colon S^2/\sim_{\mathcal F(f)}\to S^2/\sim_{\mathcal F(f)}$$2 indexes Sierpiński small spheres that remain as spheres. The resulting quotient

$$\bar f\colon S^2/\sim_{\mathcal F(f)}\to S^2/\sim_{\mathcal F(f)}$$3

produces a finite cactoid (Dudko et al., 2022).

Iterating the cactoid correspondence and passing to an inverse limit yields a totally topologically expanding map

$$\bar f\colon S^2/\sim_{\mathcal F(f)}\to S^2/\sim_{\mathcal F(f)}$$4

the maximal expanding cactoid. Under this quotient, crochet maps collapse to points, Sierpiński maps become small spheres, and bicycles become segments (Dudko et al., 2022).

The examples in the paper sharpen the interpretation. Polynomials, Newton maps, and certain matings where one polynomial has a zero-entropy Hubbard tree are examples of crochet maps. A map with Julia set a Sierpiński carpet is a Sierpiński map. Another example has quotient $$\bar f\colon S^2/\sim_{\mathcal F(f)}\to S^2/\sim_{\mathcal F(f)}$$5 equal to a segment, with quotient dynamics like a Chebyshev polynomial. These examples show that points, spheres, and segments in the cactoid encode crochet pieces, Sierpiński pieces, and annular bicycle structure, respectively (Dudko et al., 2022).

6. Related decomposition paradigms for Sierpiński-type structures

In other areas of the arXiv literature, several papers give rigorous recursive or symbolic decompositions of Sierpiński-type objects, although they do not introduce the term Canonical Crochet–Sierpiński Decomposition. One line of work treats the Sierpiński triangle through recursively defined Sierpiński matrices

$$\bar f\colon S^2/\sim_{\mathcal F(f)}\to S^2/\sim_{\mathcal F(f)}$$6

together with the entrywise rule based on the sum-of-digits function and the carry-free condition, and the semigroup identity

$$\bar f\colon S^2/\sim_{\mathcal F(f)}\to S^2/\sim_{\mathcal F(f)}$$7

which encodes a digital version of the binomial theorem (Nguyen, 2014).

A different arithmetic formulation studies base-$$\bar f\colon S^2/\sim_{\mathcal F(f)}\to S^2/\sim_{\mathcal F(f)}$$8 triangle fractals and balanced-digit hexagon fractals via no-carry digitwise addition. In that setting,

$$\bar f\colon S^2/\sim_{\mathcal F(f)}\to S^2/\sim_{\mathcal F(f)}$$9

is characterized by the condition

$\mathcal F(f)\neq\emptyset$0

and the recursive construction implies a canonical self-similar decomposition into scaled copies indexed by admissible digit pairs (Carli et al., 25 Jun 2025).

Graph-theoretic approximating models use a checked triangular generator pattern $\mathcal F(f)\neq\emptyset$1, with $\mathcal F(f)\neq\emptyset$2 dark tiles, and build $\mathcal F(f)\neq\emptyset$3 by recursively replacing each dark tile with a scaled copy of $\mathcal F(f)\neq\emptyset$4. The associated Overall Graph $\mathcal F(f)\neq\emptyset$5 and Inscribed Graph $\mathcal F(f)\neq\emptyset$6 give boundary-based and centroid-based decompositions of generalized Sierpiński gaskets (Kaszanyitzky, 2017).

Category-theoretic work on the Sierpiński carpet defines an endofunctor $\mathcal F(f)\neq\emptyset$7 that glues eight $\mathcal F(f)\neq\emptyset$8-scaled copies of a square-like object according to the $\mathcal F(f)\neq\emptyset$9 missing-center pattern, and proves that the resulting final coalgebra $S^2/\sim_{\mathcal F(f)}$0 is bi-Lipschitz equivalent to the classical carpet. In that formulation, the self-similarity equation is expressed as a final-coalgebra structure

$S^2/\sim_{\mathcal F(f)}$1

or, after completion, as an inverse isomorphism $S^2/\sim_{\mathcal F(f)}$2 (Noquez et al., 2021).

Substitution-based work on the classical Sierpiński triangle analyzes the supertiles $S^2/\sim_{\mathcal F(f)}$3 through the rule that $S^2/\sim_{\mathcal F(f)}$4 is made from three copies of $S^2/\sim_{\mathcal F(f)}$5 in the three corner positions, and refines this by a canonical decomposition of triangular pattern classes into four offset families

$S^2/\sim_{\mathcal F(f)}$6

which supports exact pattern-complexity formulas (Nilsson, 7 Oct 2025).

Recent interpolation results retain the classical Sierpiński symbolic quotient across affine-projective and Möbius deformations. In that setting, the symbolic quotient remains the classical Sierpiński quotient

$S^2/\sim_{\mathcal F(f)}$7

throughout the affine-projective family, and the paper constructs canonical homeomorphisms by matching addresses (Espigule, 22 Jun 2026).

These related formulations suggest that the phrase Canonical Crochet–Sierpiński Decomposition names a specific decomposition theory in complex dynamics—centered on Fatou adjacency, invariant multicurves, and maximal expanding quotients—while adjacent literatures use analogous recursive, symbolic, graph-theoretic, arithmetic, and coalgebraic decompositions for Sierpiński-type structures (Dudko et al., 2022).