We prove that if a primitive and non-periodic substitution is injective on initial letters, constant on final letters, and has Pisot inflation, then the R-action on the corresponding tiling space has pure discrete spectrum. As a consequence, all beta
-substitutions for beta a Pisot simple Parry number have tiling dynamical systems with pure discrete spectrum, as do the Pisot systems arising, for example, from the Jacobi-Perron and Brun continued fraction expansions.
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 tran
slation 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.
