Neighbors of self-affine tiles and Rauzy Fractals

TL;DR

This paper develops an efficient algorithm to construct neighbor graphs from contact graphs for self-affine tiles and Rauzy fractals, enhancing understanding of their tiling overlaps.

Contribution

It introduces a simplified, extendable algorithm for neighbor graph construction, improving efficiency over previous methods.

Findings

The algorithm effectively constructs neighbor graphs from contact graphs.

It extends known methods from self-affine tiles to Rauzy fractals.

The approach improves computational efficiency for analyzing tiling overlaps.

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 na\"ive algorithms for the construction of the neighbor graph.