Sierpiński Carpet Hyperbolic Components

• Sierpiński carpet hyperbolic components are defined as regions in rational moduli space or group boundaries where maps or groups yield Julia sets or boundaries homeomorphic to the standard Sierpiński carpet.
• In rational dynamics, disjoint-type components admit a global multiplier parameterization that ensures bounded, locally connected parameter spaces and robust dynamical behavior.
• Re-peripheralization in relatively hyperbolic groups transforms spherical Bowditch boundaries into carpet structures via controlled blowup models, highlighting topological rigidity.

Sierpiński carpet hyperbolic components arise in several adjacent literatures. In rational dynamics, a Sierpiński carpet hyperbolic component is a hyperbolic component in marked moduli space such that every map in the component has Julia set homeomorphic to a Sierpiński carpet; in disjoint type, such components admit a global multiplier parameterization and a strong boundedness theory (Dudko et al., 30 Sep 2025). In relatively hyperbolic group theory, a different but closely related phenomenon occurs: changing the peripheral structure of a relatively hyperbolic pair can transform a Bowditch boundary homeomorphic to $S^n$ into an $(n-1)$-dimensional Sierpiński carpet, with the peripheral circles or spheres coming from hyperbolic boundaries of removed peripheral subgroups (Souza, 2022). A third strand, centered on postcritically finite rational maps, identifies Sierpiński carpet Julia sets with Thurston-equivalence classes of expanding Thurston maps, providing a combinatorial model for the carpet locus in holomorphic dynamics (Gao et al., 2015).

1. Terminological scope and basic objects

In the parameter-space sense, the ambient setting is the fixed-point marked moduli space $\mathcal M_{d,fm}$ of degree-$d$ rational maps with ordered fixed points. A rational map is hyperbolic if every critical point converges under iteration to an attracting periodic cycle, and a hyperbolic component is a connected component of the hyperbolic locus. A hyperbolic component is of disjoint type if each grand orbit of a Fatou component contains a unique critical orbit; equivalently, every map in the component has exactly $2d-2$ distinct attracting periodic cycles. A Sierpiński carpet hyperbolic component is then a hyperbolic component $\mathcal H\subset \mathcal M_{d,fm}$ such that for every $[f]\in\mathcal H$, the Julia set $J(f)$ is a Sierpiński carpet (Dudko et al., 30 Sep 2025).

The same phrase has a broader descriptive use outside moduli theory. In relatively hyperbolic group theory, the relevant compactum is the Bowditch boundary $\partial_B(G,\mathcal P)$ of a relatively hyperbolic pair $(G,\mathcal P)$, defined via a minimal convergence action whose limit set has only conical and bounded parabolic points. In that setting, the Sierpiński carpet is not a hyperbolic component of parameter space but the boundary object produced by re-peripheralization, and its complementary “holes” correspond to embedded copies of $(n-1)$0 for certain hyperbolic peripheral subgroups $(n-1)$1 (Souza, 2022).

A planar Sierpiński carpet also appears in the classical complex-dynamical sense. For a postcritically finite rational map $(n-1)$2 with non-empty Fatou set, the Julia set $(n-1)$3 is homeomorphic to the standard Sierpiński carpet exactly when $(n-1)$4 is Thurston equivalent to an expanding Thurston map. This characterizes the carpet locus by a purely topological and combinatorial model (Gao et al., 2015).

2. Relatively hyperbolic groups and Bowditch carpet boundaries

The group-theoretic construction begins with a relatively hyperbolic pair $(n-1)$5 whose Bowditch boundary is homeomorphic to $(n-1)$6. De Souza’s main theorem states that if $(n-1)$7 is a proper subset, every subgroup in $(n-1)$8 is hyperbolic, and the dimension-dependent hypotheses of Theorem 2.8 are satisfied, then the Bowditch boundary of $(n-1)$9 with respect to $\mathcal M_{d,fm}$0 is homeomorphic to the $\mathcal M_{d,fm}$1-dimensional Sierpiński carpet (Souza, 2022).

This is a re-peripheralization theorem. One starts with $\mathcal M_{d,fm}$2, removes some peripheral subgroups from the peripheral structure, and obtains a new relatively hyperbolic structure $\mathcal M_{d,fm}$3. Under the hypotheses, the new boundary is no longer an $\mathcal M_{d,fm}$4-sphere but an $\mathcal M_{d,fm}$5-dimensional Sierpiński carpet. The mechanism relies on the blowup formalism of Proposition 1.17: if each parabolic fiber carries an appropriate minimal hyperbolic or relatively hyperbolic action, then the blown-up compactum is again a Bowditch boundary for the refined peripheral structure.

The role of the bounded parabolic points is decisive. A point $\mathcal M_{d,fm}$6 is bounded parabolic when the action of its stabilizer on $\mathcal M_{d,fm}$7 is properly discontinuous and cocompact. These are exactly the points whose stabilizers furnish the peripheral subgroups in the Bowditch boundary model. The passage from $\mathcal M_{d,fm}$8 to $\mathcal M_{d,fm}$9 replaces certain such points by compacta carrying the hyperbolic boundaries of the removed peripheral groups, and the resulting global compactum is the new Bowditch boundary.

The low-dimensional hypotheses in Theorem 2.8 reflect genuine topological obstructions rather than mere technical decoration. For $d$0 no extra geometric hypotheses are required; for $d$1 the removed peripherals are assumed virtually torsion-free; for $d$2 the statement depends on a topological characterization of the 3-dimensional Sierpiński carpet as an inverse limit of 4-balls, together with torsion-free negatively curved manifold models and the Borel Conjecture; for $d$3 one assumes that every removed peripheral has hyperbolic boundary homeomorphic to $d$4. The theorem is therefore not a generic sphere-to-carpet principle but a controlled boundary-conversion result.

3. Hyperbolic boundary components and the blowup model

The topological engine is Proposition 2.2. Let $d$5 act by homeomorphisms on $d$6, let $d$7 be the set of bounded parabolic points, and let $d$8 be a blowup map. If a countable dense $d$9-invariant set of parabolic points is replaced by closed $2d-2$0-balls $2d-2$1, while all other fibers remain trivial, then $2d-2$2 is homeomorphic to the $2d-2$3-dimensional Sierpiński carpet. This uses inverse-limit characterizations of higher-dimensional Sierpiński carpets due to Tshishiku–Walsh for $2d-2$4 and de Souza for $2d-2$5 (Souza, 2022).

In this framework, the “hyperbolic components” of the carpet boundary are the boundaries $2d-2$6 for $2d-2$7. Each such boundary is homeomorphic to $2d-2$8 and appears as the boundary of a blown-up $2d-2$9-ball $\mathcal H\subset \mathcal M_{d,fm}$0. The interiors of these balls are the complementary components of the resulting carpet, while their boundaries form the peripheral circles or spheres of the carpet. This interpretation is explicit in the proof of Theorem 2.8: the compactification of $\mathcal H\subset \mathcal M_{d,fm}$1 by $\mathcal H\subset \mathcal M_{d,fm}$2 yields a closed $\mathcal H\subset \mathcal M_{d,fm}$3-ball in each dimension handled by the theorem.

The dimension-by-dimension analysis is part of the structure. For $\mathcal H\subset \mathcal M_{d,fm}$4, the compactification is a closed interval; for $\mathcal H\subset \mathcal M_{d,fm}$5, a hyperbolic surface compactification gives a closed 2-ball; for $\mathcal H\subset \mathcal M_{d,fm}$6, 3-manifold arguments produce $\mathcal H\subset \mathcal M_{d,fm}$7 and a 3-ball; for $\mathcal H\subset \mathcal M_{d,fm}$8, the argument uses the Borel Conjecture and negative curvature; for $\mathcal H\subset \mathcal M_{d,fm}$9, Bestvina–Mess theory, EZ-structures, and Ferry’s theorem imply that an AR-space with boundary $[f]\in\mathcal H$0 and interior $[f]\in\mathcal H$1 is a closed $[f]\in\mathcal H$2-ball.

An important reverse example also appears. If $[f]\in\mathcal H$3 is a compact real hyperbolic $[f]\in\mathcal H$4-manifold with non-empty totally geodesic boundary and $[f]\in\mathcal H$5, then $[f]\in\mathcal H$6 is hyperbolic and $[f]\in\mathcal H$7 is an $[f]\in\mathcal H$8-dimensional Sierpiński carpet. Taking $[f]\in\mathcal H$9 to be the stabilizers of the peripheral $J(f)$0-spheres in that carpet, $J(f)$1 becomes relatively hyperbolic and collapsing those spheres produces a Bowditch boundary homeomorphic to $J(f)$2. Theorem 2.8 is the inverse operation: it blows an $J(f)$3 Bowditch boundary back up to a Sierpiński carpet.

4. Disjoint-type hyperbolic components in rational moduli

For rational maps, the most precise current theory concerns disjoint-type Sierpiński carpet hyperbolic components. If $J(f)$4 is a disjoint-type hyperbolic component, the attracting cycles can be labeled $J(f)$5, with multipliers $J(f)$6, and the multiplier map gives a holomorphic identification

$J(f)$7

Dudko and Luo prove that if $J(f)$8 is a Sierpiński carpet hyperbolic component of disjoint type, then $J(f)$9 is bounded in $\partial_B(G,\mathcal P)$0, and the multiplier identification extends naturally to a homeomorphism

$\partial_B(G,\mathcal P)$1

In particular, $\partial_B(G,\mathcal P)$2 is locally connected (Dudko et al., 30 Sep 2025).

The boundedness statement is parameter-space content, not merely dynamical-plane content. Since $\partial_B(G,\mathcal P)$3 is an affine algebraic variety, bounded means that the closure of $\partial_B(G,\mathcal P)$4 in $\partial_B(G,\mathcal P)$5 is compact. The theorem therefore rules out escape to infinity for this class of components and confirms, in disjoint type, the conjectural boundedness picture mentioned by McMullen.

The dynamical-plane counterpart is equally rigid. There exists $\partial_B(G,\mathcal P)$6 such that for any $\partial_B(G,\mathcal P)$7 and any non-repelling periodic point $\partial_B(G,\mathcal P)$8 of period $\partial_B(G,\mathcal P)$9, there is a quadratic-like restriction

$(G,\mathcal P)$0

with $(G,\mathcal P)$1 and $(G,\mathcal P)$2. Straightening all such restrictions yields a homeomorphism

$(G,\mathcal P)$3

where $(G,\mathcal P)$4 is the main hyperbolic component of the Mandelbrot set. Thus the closure of a disjoint-type Sierpiński carpet hyperbolic component is modeled simultaneously by the closed multiplier polydisc and by products of the quadratic main cardioid.

The proof passes through a priori bounds for boundary maps of eventually-golden-mean type. For such maps one defines valuable-attracting domains, pseudo-Siegel disks, and the pseudo-core surface

$(G,\mathcal P)$5

Its degeneration is measured by the arc and loop quantities $(G,\mathcal P)$6 and $(G,\mathcal P)$7, built from extremal width of homotopy classes of arcs and loops. Theorem C asserts uniform control of these degenerations in terms of the component $(G,\mathcal P)$8, the pulled-off constant $(G,\mathcal P)$9, and the attracting multipliers. In the Sierpiński case, the pulled-off constant is uniformly bounded by the center’s postcritical combinatorics, which is why the parameter-space closure can be controlled uniformly.

A common overextension is to treat this boundedness theorem as a statement about all Sierpiński carpet hyperbolic components. The theorem is only stated for disjoint type. The same paper explicitly introduces an obstructed boundary at infinity for general disjoint-type components and formulates a broader conjecture describing $(n-1)$00 by rational obstruction data; the Sierpiński carpet case is presented as the non-obstructed prototype.

5. Expanding Thurston models and carpet Julia sets

A complementary classification comes from postcritically finite rational maps. Qiu, Yang, and Zhu prove that a postcritically finite rational map with non-empty Fatou set is Thurston equivalent to an expanding Thurston map if and only if its Julia set is homeomorphic to the standard Sierpiński carpet. Here expansion is defined by the mesh condition for pullbacks of a Jordan curve $(n-1)$01: $(n-1)$02 This gives a purely topological characterization of the carpet locus in the postcritically finite setting (Gao et al., 2015).

The forward implication uses a quotient construction. One collapses the closure of each Fatou component to a point under the equivalence relation

$(n-1)$03

Moore’s theorem yields a quotient sphere $(n-1)$04, and the rational map descends to a Thurston map $(n-1)$05 satisfying $(n-1)$06. The theorem records the exact preservation of critical and postcritical data under this quotient: $(n-1)$07 Expansion is then deduced by comparing tile decompositions for $(n-1)$08 and for $(n-1)$09.

The converse implication constructs a semiconjugacy from $(n-1)$10 to an expanding Thurston map and uses it to force the standard carpet topology. Every Fatou component is shown to be a Jordan domain, distinct Fatou closures are disjoint, and the diameters of Fatou components tend to zero. Combined with Whyburn’s characterization, this implies that $(n-1)$11 is a Sierpiński carpet.

The paper does not formulate parameter-space hyperbolic components directly, but it does give the structural model needed for that viewpoint. In its own formulation, postcritically finite rational maps with Sierpiński carpet Julia sets are exactly those in the Thurston-equivalence classes of expanding Thurston maps. This suggests a combinatorial description of the postcritically finite carpet locus by tile structures, postcritical portraits, and local degrees. The introduction also records the conjectural statement that the components of these rational maps are relatively compact in the space of rational functions up to Möbius conjugation.

6. Related rigidity and dynamical realizations

Several further works clarify what is and is not meant by “hyperbolic” in carpet settings. Homsomboon constructs explicit homeomorphisms on Sierpiński carpets by applying the Boroński–Oprocha inverse-limit construction to hyperbolic toral automorphisms, pseudo-Anosov maps, and $(n-1)$12-baker transformations on $(n-1)$13-Chamanara surfaces. All positive real numbers are realized as metric entropy values of dynamical systems on the carpet. At the same time, these systems do not have the Bowen specification property, and in fact fail even the approximate product property. They are therefore hyperbolic in a factor or initial-system sense, but not expansive or specification-like on the carpet itself (Homsomboon, 2022).

Haïssinsky and Pilgrim provide a different rigidity model. They construct a metric Sierpiński carpet $(n-1)$14 that is quasisymmetrically co-Hopfian, and from it a visual roughly geodesic Gromov hyperbolic space $(n-1)$15 whose boundary at infinity is a Sierpiński carpet. Every quasisymmetric embedding of $(n-1)$16 into itself is onto, and every quasi-isometric embedding of $(n-1)$17 into itself is a quasi-isometry. This is a boundary-rigidity phenomenon for a hyperbolic space with carpet boundary, not a statement about hyperbolic components in moduli space (Merenkov, 2013).

Xiao studies yet another notion of rigidity for generalized Sierpiński carpets and Bedford–McMullen carpets. For a non-degenerate Bedford–McMullen carpet, any similitude sending the carpet into itself is non-oblique, and in the multiplicatively dependent case the contraction ratio satisfies $(n-1)$18. In the self-similar case with strong separation, an oblique rotational self-embedding forces $(n-1)$19. This is a classification of self-embedding types, not of hyperbolic components, but it isolates symmetry constraints that would naturally stratify parameter spaces of carpet-type sets (Xiao, 2024).

Taken together, these results show that the phrase “Sierpiński carpet hyperbolic components” is genuinely polysemous. In one precise sense it refers to bounded disjoint-type hyperbolic components in rational moduli whose Julia sets are carpets. In another, it describes the hyperbolic boundaries of peripheral subgroups that become the peripheral circles or spheres of Bowditch carpet boundaries. In adjacent work it also marks a regime where carpet dynamics inherit entropy, symbolic coding, or rigidity from hyperbolic models without becoming uniformly hyperbolic on the carpet itself.