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arXiv · 1004.2281

Exact Regularity and the Cohomology of Tiling Spaces

Lorenzo Sadun
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Abstract

The Exact Regularity Property was introduced recently as a property of homological Pisot substitutions in one dimension. In this paper, we consider exact regularity for arbitrary tiling spaces. Let ${T}$ be a $d$ dimensional repetitive tiling, and let $Ω_{T}$ be its hull. If $\check H^d(Ω_{T}, Q) = Q^k$, then there exist $k$ patches whose appearance govern the number of appearances of every other patch. This gives uniform estimates on the convergence of all patch frequencies to the ergodic limit. If the tiling ${T}$ comes from a substitution, then we can quantify that convergence rate. If ${T}$ is also one-dimensional, we put constraints on the measure of any cylinder set in $Ω_{T}$.

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Lorenzo Sadun. 2018-07-06. Exact Regularity and the Cohomology of Tiling Spaces. https://arxiv.org/abs/1004.2281

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