Cut and project sets with polytopal window I: Complexity

Henna Koivusalo, James J Walton

Research output: Contribution to journalArticle (Academic Journal)peer-review

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Abstract

We calculate the growth rate of the complexity function for polytopal cut and project sets. This generalizes work of Julien where the almost canonical condition is assumed. The analysis of polytopal cut and project sets has often relied on being able to replace acceptance domains of patterns by so-called cut regions. Our results correct mistakes in the literature where these two notions are incorrectly identified. One may only relate acceptance domains and cut regions when additional conditions on the cut and project set hold. We find a natural condition, called the quasicanonical condition, guaranteeing this property and demonstrate by counterexample that the almost canonical condition is not sufficient for this. We also discuss the relevance of this condition for the current techniques used to study the algebraic topology of polytopal cut and project sets.
Original languageEnglish
Number of pages33
JournalErgodic Theory Dynamical Systems
Early online date20 Feb 2020
DOIs
Publication statusE-pub ahead of print - 20 Feb 2020

Keywords

  • aperiodic order
  • cut and project
  • model sets
  • complexity

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