Neighbors of self-affine tiles and Rauzy Fractals

TL;DR

This work develops and simplifies algorithms to extract neighbor graphs from contact graphs for both self-affine tiles and Rauzy fractals, enabling efficient characterization of overlaps in tilings. By introducing and leveraging the $R$-corona for self-affine tiles and the $C$-Corona for Rauzy fractals, the authors provide finite-term procedures that circumvent naïve combinatorial blow-up and extend foundational results from the self-affine setting to the Rauzy context via dual substitutions and graph-directed IFS. They formalize the self-replicating boundary and contact graphs, establish correspondences between simple and full graph variants, and prove termination of the proposed algorithms. The results contribute to a deeper understanding of tiling overlaps, with potential implications for the Pisot conjecture and topological properties of Rauzy fractals in higher dimensions. The examples with explicit substitutions demonstrate practical computation of neighbor structures and validate the efficiency gains of the new methods.

Abstract

Although the theory of self-affine tiles and the theory of Rauzy fractals are quite different from each other, they have some common features. Both, self-affine tiles and Rauzy fractals have tiling properties and these tiling properties can be checked and described by certain graphs, so-called {\it contact graphs} and {\it neighbor graphs}. The contact graph is often quite easy to construct, but only the neighbor graph contains full information on the overlaps of the tiles in the presumed tiling. In the present paper we establish an algorithm that allows to construct the neighbor graph starting from the contact graph. Such an algorithm is already known in the case of self-affine tiles. In the present paper we give a simplified proof of this algorithm that can be extended to the case of Rauzy fractals. Our algorithms are more efficient than naïve algorithms for the construction of the neighbor graph.

Figures (11)

• Figure 1: Knuth's twindragon, a famous self-affine tile (see e.g.Knuth:98), and a three-dimensional self-affine tile.
• Figure 2: In the 3rd approximation one can jump from $\mathcal{T}_3$ via $\mathcal{T}_3+\bm^{(1)}$ to $\mathcal{T}_3+\bm^{(1)}+\bm^{(2)}$; the tile $\mathcal{T}_3+\bm^{(1)}+\bm^{(2)}$ is not a contact neighbor of $\mathcal{T}_3$ (left panel). However, $\mathcal{T}+\bm^{(1)}+\bm^{(2)}$ is a neighbor of $\mathcal{T}$. In the right panel we can see that $\mathcal{T}+\bm^{(1)}+\bm^{(2)}$ is a neighbor of $\mathcal{T}$, i.e., $\mathcal{T} \cap (\mathcal{T}+\bm^{(1)}+\bm^{(2)})\not=\emptyset$.
• Figure 3: Geometric meaning of the representations of the nodes in the edge $\bm_{k-1}^{(1)}+\bm_{k-1}^{(2)}\xrightarrow{\mathbf{d}_k|\mathbf{d}'_k}\bm_{k}^{(1)}+\bm_{k}^{(2)}$. Here (1) is the subtile $M^{-1}(\mathcal{T}+\mathbf{d}_k)$ of $\mathcal{T}$, (2) is the subtile $M^{-1}(\mathcal{T}+\bm_{k}^{(1)}+\mathbf{d}_k)$ of $\mathcal{T}+\bm_{k-1}^{(1)}$ and (3) is the subtile $M^{-1}(\mathcal{T}+\bm_{k}^{(1)}+\bm_{k}^{(2)}+\mathbf{d}_k)=M^{-1}(\mathcal{T}+M(\bm_{k-1}^{(1)}+\bm_{k-1}^{(2)})+\mathbf{d}'_k)$ of $\mathcal{T}+\bm_{k-1}^{(1)}+\bm_{k-1}^{(2)}$. One can see that if the small tiles are contact neighbors, then also the large tiles are contact neighbors (Drawn for the 3rd approximation.)
• Figure 4: Prefix-suffix graph for $\sigma_1$ (left) and $\sigma_2$ (right).
• Figure 5: Rauzy fractals and their subtiles associated with $\sigma_1$ (left) and $\sigma_2$ (right).

Theorems & Definitions (38)

• Lemma 1.1
• Definition 1.2: $R$-corona
• Lemma 1.3
• proof
• Proposition 1.4
• proof
• Theorem 1.5
• proof
• Definition 2.1: Prefix-suffix graph; cf. e.g. Canterini-Siegel:01a
• Example 2.2: Prefix-suffix graph