Factors of Pisot tiling spaces and the coincidence rank conjecture

TL;DR

This paper investigates the structure of Pisot substitution tiling spaces, focusing on those with non-pure discrete spectrum, and establishes bounds related to the Coincidence Rank Conjecture for specific cases.

Contribution

It introduces a lower bound on cohomology for certain Pisot tiling spaces and proves the Coincidence Rank Conjecture for rank two cases.

Findings

Established a lower bound on cohomology for specific tiling spaces.

Proved the Coincidence Rank Conjecture for coincidence rank two.

Linked the structure of tiling spaces to spectral properties.

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.