Exact Regularity and the Cohomology of Tiling Spaces

Details Exact Regularity and the Cohomology of Tiling Spaces Exact Regularity and the Cohomology of Tiling Spaces

2010-04-13

arxiv.org/abs/1004.2281v1

Mathematics

Dynamical Systems

15 pages, including 3 figures

Scientific paper

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 $\bT$ be a $d$ dimensional repetitive tiling, and let $\Omega_{\bT}$ be its hull. If $\check H^d(\Omega_{\bT}, \Q) = \Q^k$, 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 $\bT$ comes from a substitution, then we can quantify that convergence rate. If $\bT$ is also one-dimensional, we put constraints on the measure of any cylinder set in $\Omega_{\bT}$.

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