no. 2
Factors of Pisot tiling spaces and the Coincidence Rank Conjecture
[Facteurs de l'espace des pavages d'une substitution de Pisot et conjecture du rang de coïncidence]
Bulletin de la Société Mathématique de France, Tome 143 (2015) no. 2, pp. 357-381

We consider the structure of Pisot substitution tiling spaces, in particular, the structure of those spaces for which the translation action does not have pure discrete spectrum. Such a space is always a measurable m-to-one cover of an action by translation on a group called the maximal equicontinuous factor. The integer m is the coincidence rank of the substitution and equals one if and only if translation on the tiling space has pure discrete spectrum. By considering factors intermediate between a tiling space and its maximal equicontinuous factor, we establish a lower bound on the cohomology of a one-dimensional Pisot substitution tiling space with coincidence rank two and dilation of odd norm. The Coincidence Rank Conjecture, for coincidence rank two, is a corollary.

Nous considérons la structure de l'espace des pavages d'une substitution de Pisot, en particulier dans le cas où l'action par les translations n'a pas de spectre purement discret. Un tel espace est toujours recouvrement presque partout de degré m d'une translation sur un groupe. Ce groupe s'appelle le facteur maximal équicontinu. L'entier m est le rang de coïncidence de la substitution et il vaut 1 si et seulement si l'action par les translations a un spectre purement discret. En tenant compte des facteurs intermédiaires entre l'espace de pavage et son facteur maximal équicontinu, nous établissons une borne inférieure sur la cohomologie des espaces des pavages d'une substitution de Pisot unidimensionelle avec rang de coïncidence 2 et dilatation de norme impaire. La conjecture du rang de coïncidence, pour un rang de coïncidence égal a 2, en découle en tant que corollaire.

Publié le :
DOI : 10.24033/bsmf.2691
Classification : 37B05, 37B50
Keywords: Pisot substitution, tiling space
Mots-clés : Substitution de Pisot, espace des pavages 
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Barge, Marcy. Factors of Pisot tiling spaces and the Coincidence Rank Conjecture. Bulletin de la Société Mathématique de France, Tome 143 (2015) no. 2, pp. 357-381. doi: 10.24033/bsmf.2691

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