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arXiv · 1301.7094

Factors of Pisot tiling spaces and the coincidence rank conjecture

Marcy Barge
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Abstract

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 tillng space with coincidence rank two and dilation of odd norm. The Coincidence Rank Conjecture, for coincidence rank two, is a corollary.

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Marcy Barge. 2013-01-29. Factors of Pisot tiling spaces and the coincidence rank conjecture. https://arxiv.org/abs/1301.7094

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