Maximal Expanding Cactoid in Dynamics

• Maximal expanding cactoids are canonical quotients of Böttcher expanding maps that collapse Fatou components obstructing full expansion.
• They classify dynamics into crochet, Sierpiński, and mixed cases through a canonical decomposition that preserves the core expanding behavior.
• The construction uses crochet decomposition and inverse-limit techniques to ensure every semiconjugacy factors through the maximal totally expanding quotient.

A maximal expanding cactoid is, in the formal sense used in holomorphic and topological dynamics, the canonical quotient of a Böttcher expanding map with non-empty Fatou set, obtained by collapsing exactly the Fatou structure that obstructs total expansion while retaining the expanding part of the dynamics on a cactoid. The quotient space is $S^2/\!\sim_{\mathcal F(f)}$, where $\sim_{\mathcal F(f)}$ is the smallest closed equivalence relation on $S^2$ generated by identifying all points in every Fatou component, and the induced map on the quotient is the maximal totally expanding quotient of the original system (Dudko et al., 2022). In the paper that introduced the term, the construction is applied to postcritically finite rational maps and, more generally, to Böttcher expanding maps; elsewhere, the phrase “maximal” combined with “cactus” or “cactoid” appears only informally in graph-theoretic extremal problems, where it refers to extremality for spectral, distance-spectral, Sombor, Kirchhoff, or density objectives rather than to cactoid dynamics (Mallik, 2011, Liu, 2021, Wang et al., 2023, Wang et al., 2015, Zhang et al., 4 Jun 2026).

1. Definition and formal setting

For a Böttcher expanding map $f\colon (S^2,A)$ with non-empty Fatou set $\mathcal F(f)\neq\emptyset$, the equivalence relation $\sim_{\mathcal F(f)}$ is defined as the smallest closed equivalence relation on $S^2$ generated by identifying all points in every Fatou component. The quotient

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

is a cactoid, and $f$ descends to a map

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

The main theorem states that this induced map is the maximal totally expanding quotient: any other semi-conjugacy from $\sim_{\mathcal F(f)}$0 to a totally topologically expanding map factors through the quotient map $\sim_{\mathcal F(f)}$1 (Dudko et al., 2022).

A cactoid is recalled in that work as a continuous monotone image of $\sim_{\mathcal F(f)}$2. Topologically, such a space is described as a locally connected continuum composed of countably many spheres and segments pairwise intersecting in at most one point. The maximal expanding cactoid is therefore not merely a quotient space; it is a quotient endowed with an induced totally topologically expanding dynamics (Dudko et al., 2022).

The adjective “maximal” has a universal property. It does not mean maximal entropy, maximal measure, or maximal graph density. It means that among quotients of the original system that are totally topologically expanding, this quotient is terminal: every such quotient factors through it. A plausible implication is that the maximal expanding cactoid should be regarded as the largest canonically defined expansion-preserving collapse of the original dynamics.

2. Böttcher expanding maps and the role of expansion

The construction is formulated for Böttcher expanding maps, which the paper presents as metric models of postcritically finite rational maps. A Thurston map $\sim_{\mathcal F(f)}$3 is metrically expanding rel. $\sim_{\mathcal F(f)}$4 if there exists a length metric on $\sim_{\mathcal F(f)}$5 such that every nontrivial rectifiable curve has lifts under $\sim_{\mathcal F(f)}$6 of strictly smaller length and $\sim_{\mathcal F(f)}$7 has Böttcher normalization at points of $\sim_{\mathcal F(f)}$8: the first return map is locally conjugate to

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

If $S^2$0, the map is Böttcher expanding (Dudko et al., 2022).

The quotient cactoid carries a different but related expansion notion. A partial self-cover $S^2$1 is topologically expanding if there exists a finite open cover $S^2$2 such that the pullback covers satisfy

$S^2$3

If $S^2$4, the map is totally topologically expanding. The induced quotient map $S^2$5 on the cactoid is of this latter type (Dudko et al., 2022).

This distinction is structurally important. The original map lives on a sphere and may fail to be globally expanding because Fatou components are non-expanding regions. The quotient removes precisely those regions, up to the closure constraints encoded in $S^2$6, and the result is a totally expanding dynamical system on a cactoid rather than on a sphere. This suggests that the cactoid is the natural ambient space once one insists on retaining only the expanding core.

3. Crochet decomposition and the anatomy of the quotient

The maximal expanding cactoid is organized by the paper’s canonical “crochet decomposition.” The two principal building blocks are Sierpiński maps and crochet maps. A Böttcher expanding map is Sierpiński if its Julia set is homeomorphic to the standard Sierpiński carpet; equivalently, $S^2$7, Fatou components have pairwise disjoint closures, and the closure of every Fatou component is a Jordan domain. A Böttcher expanding map is crochet if there exists a connected forward-invariant zero-entropy graph $S^2$8 containing $S^2$9 (Dudko et al., 2022).

The canonical decomposition theorem states that there is a unique canonical invariant multicurve $f\colon (S^2,A)$0 whose small maps are Sierpiński and crochet maps such that, for the quotient map

$f\colon (S^2,A)$1

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

The same paper gives an alternative description in terms of bicycles. Maximal Sierpiński small maps are well-defined; if two bicycles have positive geometric intersection, they lie within a small Sierpiński map; and the crochet multicurve $f\colon (S^2,A)$2 is generated by boundaries of maximal Sierpiński small maps together with the remaining bicycles (Dudko et al., 2022).

These statements identify the local models of the quotient. Small Sierpiński maps become small spheres; small crochet maps collapse to points; bicycles correspond to arcs or segments. A plausible implication is that the cactoid records exactly which parts of the Julia–Fatou decomposition remain sphere-like and which degenerate to lower-dimensional connectors.

4. Construction of the quotient cactoid

The intrinsic quotient is defined directly by the Fatou relation $f\colon (S^2,A)$3, but the paper also gives an explicit quotient procedure from decomposition data. One writes

$f\colon (S^2,A)$4

where $f\colon (S^2,A)$5 indexes small spheres, $f\colon (S^2,A)$6 indexes crochet spheres, $f\colon (S^2,A)$7 indexes Sierpiński spheres, $f\colon (S^2,A)$8 is the submulticurve generated by bicycles, and $f\colon (S^2,A)$9. The quotient then proceeds by collapsing crochet curves to points, collapsing bicycle annuli to closed segments by collapsing each foliation curve $\mathcal F(f)\neq\emptyset$0 to a point, and collapsing crochet small spheres to points. The resulting map is denoted

$\mathcal F(f)\neq\emptyset$1

(Dudko et al., 2022).

The quotient fits into a commutative diagram with induced maps

$\mathcal F(f)\neq\emptyset$2

and a compatible forgetful monotone map

$\mathcal F(f)\neq\emptyset$3

The maximal cactoid map is then obtained as the inverse limit of the correspondence

$\mathcal F(f)\neq\emptyset$4

producing $\mathcal F(f)\neq\emptyset$5 and an induced map $\mathcal F(f)\neq\emptyset$6. The semiconjugacy $\mathcal F(f)\neq\emptyset$7 satisfies

$\mathcal F(f)\neq\emptyset$8

and, after returning from the formal model to the original Böttcher expanding map, one obtains

$\mathcal F(f)\neq\emptyset$9

(Dudko et al., 2022).

The final identification theorem states

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

Thus the explicit inverse-limit construction and the intrinsic quotient by Fatou components produce the same cactoid (Dudko et al., 2022).

5. Special cases and characterizations

The theory isolates several extremal quotient types. A Böttcher expanding map is crochet if and only if the quotient cactoid is a singleton. It is Sierpiński if and only if $\sim_{\mathcal F(f)}$1 is a sphere and $\sim_{\mathcal F(f)}$2 is trivial on $\sim_{\mathcal F(f)}$3. A map is Sierpiński-free if and only if no decomposition produces a Sierpiński small map, if and only if no small map in the crochet decomposition is Sierpiński, if and only if $\sim_{\mathcal F(f)}$4 is a dendrite (Dudko et al., 2022).

These cases can be organized succinctly:

Dynamical type	Quotient $\sim_{\mathcal F(f)}$5	Structural outcome
Crochet map	Singleton	All relevant Fatou connectivity collapses
Sierpiński map	Sphere	Fatou components collapse without nontrivial identifications on $\sim_{\mathcal F(f)}$6
Sierpiński-free map	Dendrite	No small Sierpiński pieces survive
Mixed case	Genuine cactoid	Spheres, segments, and points coexist

The paper’s examples illustrate these regimes. Crochet examples include polynomials, Newton maps, and certain matings where one polynomial has a zero-entropy Hubbard tree; in these cases any Fatou component can be connected to another by a countable chain of touching Fatou components, so the quotient is a singleton. In the Sierpiński-carpet case, the quotient is a sphere. In a tuning example, the map decomposes canonically into a crochet piece and a Sierpiński piece, and the quotient is still a sphere but the equivalence relation is not trivial on the postcritical set. In a segment or Chebyshev-type example, the quotient is a segment, with quotient dynamics of a Chebyshev polynomial (Dudko et al., 2022).

A common misconception is that collapsing each Fatou component individually is all that occurs. The formal definition uses the smallest closed equivalence relation generated by those collapses, so limit configurations of touching Fatou components can force additional identifications. This is precisely why the quotient need not remain a sphere.

6. Maximality, computability, and relation to other “expanding cactus” usages

The universal property of the maximal expanding cactoid is proved by showing that any semiconjugacy $\sim_{\mathcal F(f)}$7 from $\sim_{\mathcal F(f)}$8 to a totally topologically expanding map $\sim_{\mathcal F(f)}$9 must collapse each periodic Fatou component, hence every Fatou component. Therefore $S^2$0 respects $S^2$1 and factors through $S^2$2. In this precise sense, the quotient is maximal (Dudko et al., 2022).

The construction is also effective. The paper gives a Crochet Algorithm: compute maximal clusters of touching Fatou components and their boundary multicurve; decompose along this multicurve; repeat on small maps until all are crochet or Sierpiński; glue small crochet maps corresponding to the same point in the quotient. The authors state that all steps can be performed symbolically from a sphere biset input (Dudko et al., 2022).

The phrase “maximal expanding cactoid” should be distinguished from several graph-theoretic extremal usages of “maximal cactus.” In graph spectral theory, among all cactus graphs on $S^2$3 vertices, the unique graph with maximum adjacency spectral radius is $S^2$4, obtained from the star $S^2$5 by adding $S^2$6 independent edges between disjoint pairs of pendant vertices (Mallik, 2011). For the Sombor index, the unique maximizers are hub-centered cacti $S^2$7 and $S^2$8, again concentrating triangles and attachments at a dominant hub (Liu, 2021). For the distance spectral radius with fixed order and number of cycles, the unique maximizer is

$S^2$9

a balanced saw-graph (Wang et al., 2023). For the Kirchhoff index, the unique maximizer is instead the chain cactus $S^2/\!\sim_{\mathcal F(f)}$0, built from two terminal chains of triangles joined by a path (Wang et al., 2015). For generalized $S^2/\!\sim_{\mathcal F(f)}$1-cacti, recent work treats maximality as an edge-density problem under the condition that each edge lies on at most $S^2/\!\sim_{\mathcal F(f)}$2 cycles (Zhang et al., 4 Jun 2026).

These graph-theoretic results are conceptually related only at the level of terminology. They do not define a cactoid as a monotone image of $S^2/\!\sim_{\mathcal F(f)}$3, and they do not address totally topologically expanding quotient dynamics. In the dynamical literature, the maximal expanding cactoid is a canonical quotient object attached to postcritically finite dynamics; in graph theory, “maximal cactus” refers to an extremal graph for a chosen invariant.

7. Conceptual significance

Within the theory of postcritically finite rational maps and Böttcher expanding maps, the maximal expanding cactoid packages the expanding core of the dynamics in a canonical topological model. Small Sierpiński maps remain visible as spheres, crochet pieces collapse to points, and bicycles become segments; the quotient is neither an arbitrary collapse nor an ad hoc simplification, but the unique quotient through which every totally expanding semiconjugate model must pass (Dudko et al., 2022).

This suggests a useful interpretation. The maximal expanding cactoid is the precise topological object obtained when one removes all non-expanding Fatou structure while preserving the full quotient dynamics compatible with total topological expansion. In that sense, it is both a decomposition theorem and a universal model: a decomposition because it resolves the original system into crochet and Sierpiński pieces, and a universal model because it carries the maximal totally expanding quotient dynamics on a cactoid (Dudko et al., 2022).