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Exact regularity and the cohomology of tiling spaces

Published online by Cambridge University Press:  18 January 2011

LORENZO SADUN
LORENZO SADUN*
Affiliation:
Department of Mathematics, University of Texas, Austin, TX 78712, USA (email: sadun@math.utexas.edu)
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Abstract

Exact regularity was introduced recently as a property of homological Pisot substitutions in one dimension. In this paper, we consider the analog of exact regularity for arbitrary tiling spaces. Let T be a d-dimensional repetitive tiling, and let Ω be its hull. If Ȟd(Ω,ℚ)=ℚk, then there exist k patches each of whose appearances governs 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 6 , December 2011 , pp. 1819 - 1834
Copyright
Copyright © Cambridge University Press 2011

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