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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2511.16442 |
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| _version_ | 1866911277464420352 |
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| author | Loridant, Benoît Thuswaldner, Jörg M. Zhang, Shu-Qin |
| author_facet | Loridant, Benoît Thuswaldner, Jörg M. Zhang, Shu-Qin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_16442 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Neighbors of self-affine tiles and Rauzy Fractals Loridant, Benoît Thuswaldner, Jörg M. Zhang, Shu-Qin Metric Geometry 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. |
| title | Neighbors of self-affine tiles and Rauzy Fractals |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2511.16442 |