A polygon carpet is a planar self-similar fractal generated by finitely many contracting similarities of a polygon satisfying the open set condition, connectivity, symmetry and non-triviality; it is perfect if any two first-generation cells are either disjoint or meet side-to-side or corner-to-corner, and bordered if the images of the similarities cover the whole boundary of the polygon, in which case the polygon is a triangle or square and the contraction ratios are allowed to be distinct, so that the fractal may be irrationally ramified (cells meeting along proper sub-segments of edges). Cao, Qiu and Wang (J. Anal. Math. 2026) proved that every perfect polygon carpet, and every hollow bordered polygon carpet satisfying the explicit geometric conditions (H) and (C-3)/(C-4), carries a standard self-similar Dirichlet form — hence a symmetric diffusion with sub-Gaussian heat kernel estimates — while Cao and Qiu (Proc. AMS 2023) constructed a bordered square carpet on which no such standard self-similar energy exists. The paper leaves open the existence problem for general polygon carpets: determine which polygon carpets admit a standard self-similar Dirichlet form; the classes not covered by Theorem 1 include bordered polygon carpets outside its sufficient conditions and polygon carpets that are neither perfect nor bordered. The open fronts are existence for polygon carpets that are neither perfect nor bordered, for non-hollow bordered carpets (first-generation cells not all on the boundary) and for hollow carpets outside the known sufficient conditions, identification of the obstruction mechanisms beyond the known corner-disconnection counterexample, and whether the loop-intersection, hollow, and connectivity hypotheses can be weakened to intrinsic geometric criteria that are both necessary and sufficient.

Background

Analysis on fractals studies diffusions and Laplacians on self-similar sets. An iterated function system (IFS) is a finite family of contracting similarities Ψi\Psi_i of a planar polygon AA; its attractor KK is the unique compact set with K=⋃iΨiKK = ⋃_i \Psi_i K. The classical example is the Sierpinski carpet (SC), obtained from the unit square by keeping the eight sub-squares of side 1/31/3 and iterating. Barlow and Bass constructed Brownian motion on the SC probabilistically and extended the construction to higher-dimensional generalized Sierpinski carpets, while Kusuoka and Zhou (Probab. Theory Related Fields 1992) introduced a complementary analytic approach, building Dirichlet forms as limits of rescaled energies on the graphs of nn-th generation cells; the equivalence of the two approaches, and the uniqueness of the resulting Brownian motion, were settled by Barlow, Bass, Kumagai and Teplyaev (J. Eur. Math. Soc. 2010). A Dirichlet form on L2(K,μ)L^2(K, μ) is a non-negative, closed, densely defined quadratic form with the Markov property; if it is local and regular it generates a symmetric diffusion. A sub-Gaussian heat kernel estimate is a two-sided bound pt(x,y)≈t−dH/βexp⁡(−c(∣x−y∣β/t)1/(β−1))p_t(x,y) ≈ t^{-d_H/β} \exp(-c(|x-y|^β/t)^{1/(β-1)}) for the transition density, where dHd_H is the Hausdorff dimension and ββ the walk dimension (the exponent with which diffusion time scales with Euclidean displacement, t≍rβt \asymp r^β). For post-critically finite (p.c.f.) self-similar sets — finitely ramified fractals whose cells intersect in finitely many points — Kigami's harmonic calculus and Sabot's algebraic theory reduce the existence of self-similar energies to finite-dimensional nonlinear eigenvalue problems. Infinitely ramified carpet-like fractals, where cells intersect along line segments, have no such reduction, and the analytic route of Kusuoka and Zhou must verify, at every scale, the comparability between Poincaré constants (measuring oscillation of functions of unit graph energy) and resistance constants (measuring effective resistance of the cell graphs, i.e., the energy-minimizing cost of separating cell blocks by boundary values 1 and 0). Cao and Qiu extended the existence theory to unconstrained Sierpinski carpets (Probab. Theory Related Fields 2024) by a purely analytic chaining of resistances. Cao, Qiu and Wang (arXiv:2206.00040, published in J. Anal. Math. 2026) then introduced polygon carpets: attractors of an IFS on a polygon AA with N0N_0 vertices satisfying the open set condition (disjoint interiors of the cells), connectivity, invariance of the cell pattern under the symmetry group of AA, and non-triviality. A polygon carpet is perfect if any two first-generation cells are either disjoint or meet side-to-side or corner-to-corner (which forces equal contraction ratios), and bordered if ∂A⊂⋃iΨiA∂A ⊂ ⋃_i Ψ_i A, the boundary-inclusion condition familiar from the SC; a bordered polygon carpet must have N0=3N_0 = 3 or 44 (triangles or squares), and distinct contraction ratios are allowed, so bordered polygon carpets include many irrationally ramified fractals — fractals whose cells meet along proper sub-segments of edges rather than only in points or along whole edges — such as Kigami's Sierpinski cross (Mem. AMS 2009). A bordered carpet is hollow if all first-generation cells lie along ∂A∂A; the paper's conditions are the hollow condition (H) — every first-generation cell meets ∂A∂A, and any two non-corner cells intersect, if at all, in a line segment — the connectivity conditions (C-3) for triangles and (C-4) for squares describing how the corner cells attach to the rest, and the intermediate loop-intersection condition (LI). The key analytic gate is condition (B), the comparability σm≤CRmσ_m ≤ C R_m for all mm, where σmσ_m bounds the squared oscillation of block averages of unit-energy functions between neighboring cell blocks and RmR_m is the minimal effective resistance between an mm-block and the far-away blocks; within the paper's framework (a Kusuoka-type scheme adapted to distinct ratios), verifying (B) yields a local, regular self-similar Dirichlet form on L2(K,μ)L^2(K, μ) (μμ the normalized Hausdorff measure) with the standard scaling E(f)=∑iρi−θE(f∘Ψi)E(f) = ∑_i ρ_i^{-θ} E(f ∘ Ψ_i) for some θθ, called a standard self-similar Dirichlet form; by Kigami's resistance-form machinery (Mem. AMS 2012) such a form yields a diffusion with sub-Gaussian heat kernel estimates. Theorem 1 of that paper proves (B), hence existence, for all perfect polygon carpets and for all bordered polygon carpets satisfying (H) together with (C-3) or (C-4). However, Cao and Qiu (Proc. AMS 2023) had already constructed a Sierpinski-carpet-like bordered square carpet — opposite sides strongly connected through large central cells while the four corner vertices are loosely connected to the center, an idea inspired by Sabot's finitely ramified examples — on which no standard self-similar Dirichlet form exists: the geometry forces contradictory bounds θ≤1/2θ ≤ 1/2 and θ>1/2θ > 1/2 on the scaling exponent. Hence existence genuinely depends on the geometry of the cell pattern, and the paper states that the existence problem for general polygon carpets is much more difficult and interesting, expressing the hope that the work inspires further investigation.

Problem Statement

Let AA be a polygon in R2\mathbb{R}^2 with N0≥3N_0 \geq 3 vertices and symmetry group GG, and let {Ψi}i∈S\{\Psi_i\}_{i \in S} be a finite family of contracting similarities of AA with ratios ρi\rho_i, satisfying the open set condition (disjoint interiors Ψi(A∘)∩Ψj(A∘)=∅\Psi_i(A^\circ) \cap \Psi_j(A^\circ) = \emptyset for i≠ji \neq j), connectivity of the attractor, invariance of the cell pattern under GG, and non-triviality (⋃iΨiA≠A\bigcup_i \Psi_i A \neq A); the attractor KK of the family is then a polygon carpet. KK is perfect if any two first-generation cells are either disjoint or meet side-to-side or corner-to-corner, and bordered if it further satisfies the boundary-inclusion condition ∂A⊂⋃iΨiA\partial A \subset \bigcup_i \Psi_i A, which forces N0=3N_0 = 3 or 44; in the bordered case the ratios ρi\rho_i may be distinct, so KK may be irrationally ramified. Call a local, regular, conservative, irreducible Dirichlet form (E,F)(E, \mathcal{F}) on L2(K,μ)L^2(K, \mu), with μ\mu the normalized Hausdorff measure, a standard self-similar Dirichlet form if for some θ∈R\theta \in \mathbb{R} it satisfies E(f)=∑i∈Sρi−θE(f∘Ψi)E(f) = \sum_{i \in S} \rho_i^{-\theta} E(f \circ \Psi_i) for all f∈Ff \in \mathcal{F} together with the standard domain condition f∘Ψi∈F∩C(K)f \circ \Psi_i \in \mathcal{F} \cap C(K) for all ii implies f∈F∩C(K)f \in \mathcal{F} \cap C(K). Known: such a form exists on every perfect polygon carpet and on every bordered polygon carpet satisfying the hollow condition (H) with (C-3) or (C-4); and there is an explicit bordered square carpet (Cao-Qiu) on which no standard self-similar Dirichlet form exists. The open problem is the existence question for general polygon carpets underlying these facts, with one common objective — decide, from geometric features of the IFS cell pattern, exactly which polygon carpets admit a standard self-similar Dirichlet form (implying a symmetric diffusion with sub-Gaussian heat kernel estimates on KK). Its concrete fronts are: (1) existence beyond the known sufficient classes, in particular for polygon carpets that are neither perfect nor bordered, for non-hollow bordered polygon carpets, whose first-generation cells are not all located on ∂A\partial A, and for hollow carpets — those with all first-generation cells on ∂A\partial A — that fail the line-segment requirement of (H) or fail (C-3)/(C-4) (or the loop-intersection condition (LI), which (H) together with (C-3)/(C-4) implies); (2) identification of the obstruction mechanisms for nonexistence — whether loose connection of boundary vertices to the interior, as in the known counterexample, is the only source of nonexistence, or whether further mechanisms exist, together with any provable necessary geometric conditions; and (3) determination of whether (H), (C-3)/(C-4), and (LI) can be weakened or replaced by intrinsic criteria that are necessary as well as sufficient. These three fronts are independent sub-questions of this single objective rather than alternative resolutions of it: the objective is the general geometric characterization — a decision of exactly which polygon carpets admit a standard self-similar Dirichlet form.

References

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