2013/01/29 by Marcy Barge, Barge, Marcy
Mathematics · #05D10 #68R15 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:05D10 #msc:68R15
paper · pdf · doi:10.48550/arxiv.1301.7094
18 pages
arxiv created 2013/01/29 · arxiv updated 2013/01/31
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.