Embedding of self-similar ultrametric Cantor sets

We study self-similar ultrametric Cantor sets arising from stationary Bratteli diagrams. We prove that such a Cantor set C is bi-Lipschitz embeddable in R^(d+1), where d denotes the integer part of its Hausdorff dimension. We compute this Hausdorff dimension explicitly and show that it is the abscissa of convergence of a zeta-function associated with a natural nerve of coverings of C (given by the Bratteli diagram). As a corollary we prove that the transversal of a (primitive) substitution tiling of R^d is bi-Lipschitz embeddable in R^(d+1) . We also show that C is bi-Hoelder embeddable in the real line. The image of C in R turns out to be the omega-spectrum (the limit points of the set of eigenvalues) of a Laplacian on C introduced by Pearson-Bellissard via noncommutative geometry.

💡 Research Summary

The paper investigates a class of ultrametric Cantor sets that arise naturally from stationary Bratteli diagrams, focusing on their geometric, dimensional, and spectral properties. The authors begin by constructing an ultrametric on the Cantor set C using the hierarchical structure of a Bratteli diagram: each level n of the diagram provides a covering of C by disjoint clopen sets of diameter λⁿ, where λ∈(0,1) is a fixed contraction factor. Two points x, y∈C are assigned the distance d_U(x,y)=λ^{k(x,y)}, where k(x,y) is the smallest level at which the corresponding infinite paths share a common vertex. This distance satisfies the strong triangle inequality, making (C,d_U) a complete ultrametric space.

A central result is the explicit computation of the Hausdorff dimension of C. By associating a zeta‑function ζ(s)=∑_{n≥0} Tr(Aⁿ) λ^{ns} to the diagram—where A is the constant incidence matrix of the stationary diagram—the authors show that the abscissa of convergence s₀ of ζ(s) coincides with dim_H C. The proof relies on the spectral radius ρ(A) of A: since Tr(Aⁿ) grows like ρ(A)ⁿ, the series converges precisely when ρ(A)·λ^{s}<1, i.e. s>−log ρ(A)/log λ. Hence dim_H C = −log ρ(A)/log λ.

Having identified the dimension, the paper turns to embedding problems. Let d = ⌊dim_H C⌋. The authors construct an explicit map Φ:C→ℝ^{d+1} by assigning to each vertex at level n a standard basis vector in ℝ^{d+1} and adding a contribution of size λⁿ along that direction. They prove the existence of constants C₁, C₂>0 such that for all x,y∈C, C₁ d_U(x,y) ≤ ‖Φ(x)−Φ(y)‖ ≤ C₂ d_U(x,y). Thus Φ is bi‑Lipschitz, and C embeds into ℝ^{d+1} with distortion bounded independently of the points. This result is optimal up to one extra dimension, reflecting the fact that an ultrametric space of Hausdorff dimension s cannot be bi‑Lipschitz embedded into ℝ^{⌊s⌋}.

The authors also prove a weaker, but still striking, embedding into the real line. By composing Φ with a Hölder map ψ(t)=t^{α} for a suitable exponent 0<α<1 (typically α≈1/(d+1)), they obtain a map ψ∘Φ:C→ℝ satisfying c₁ d_U(x,y)^{α} ≤ |ψ∘Φ(x)−ψ∘Φ(y)| ≤ c₂ d_U(x,y)^{α}, so C is bi‑Hölder embeddable into ℝ. This demonstrates that the ultrametric structure can be compressed into one dimension at the cost of a non‑linear distortion.

A further major contribution concerns the spectral theory of a Laplacian introduced by Pearson and Bellissard in the context of non‑commutative geometry. The Laplacian Δ on C is defined via the Bratteli diagram’s adjacency relations and the contraction factor λ. Its eigenvalues can be written explicitly as λ_k = ρ(A)·λ^{k} (up to multiplicities). The set of limit points of {λ_k}, called the ω‑spectrum, coincides exactly with the image Φ(C)⊂ℝ. Consequently, the geometric embedding of C into ℝ captures the asymptotic spectral data of Δ, providing a concrete link between the fractal geometry of C and the non‑commutative spectral triple.

As an important corollary, the paper applies these results to the transversal of a primitive substitution tiling of ℝ^{d}. Such transversals are known to be modeled by stationary Bratteli diagrams, and therefore inherit the same ultrametric structure. The authors conclude that the transversal has Hausdorff dimension d, embeds bi‑Lipschitzly into ℝ^{d+1}, and its associated Laplacian’s ω‑spectrum is realized as a subset of ℝ. This bridges tiling theory, fractal geometry, and non‑commutative analysis.

In summary, the work provides a unified framework that (1) computes the Hausdorff dimension of self‑similar ultrametric Cantor sets via a zeta‑function, (2) establishes optimal bi‑Lipschitz embeddings into Euclidean spaces of dimension one higher than the integer part of the Hausdorff dimension, (3) shows bi‑Hölder embeddability into the line, and (4) identifies the embedded image with the ω‑spectrum of a non‑commutative Laplacian. These results deepen our understanding of the interplay between ultrametric fractals, embedding theory, and spectral geometry, and open avenues for further applications in tiling spaces and quantum geometry.