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Homological Pisot substitutions and exact regularity. (English) Zbl 1257.37010
A circle of results and questions in dynamics and number theory are concerned with the special properties of Pisot numbers, and in many cases there is an underlying substitution system built on a finite alphabet. Here there is a standard conjecture, formulated in this paper as asking if the tiling flow associated with a one-dimensional substitution of unimodular irreducible Pisot type has pure discrete spectrum. Here the authors introduce a notion of homological Pisot substitutions, namely one-dimensional substitution tiling spaces whose dilation factor is a degree $$d$$ Pisot number with the property that the first rational Čech homology is $$d$$-dimensional. Examples of such substitutions are found with tiling flows whose spectrum is not pure discrete. The examples constructed are not unimodular, and it is conjectured here that the coincidence rank must divide a power of the norm of the dilation. Evidence is given to support this conjecture via constraints arising from the dynamics on the possible measures of cylinder sets in the corresponding tiling space.

##### MSC:
 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) 37A30 Ergodic theorems, spectral theory, Markov operators
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