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1 results

  • Cut and project sets with polytopal window I: Complexity

  • Part of
  • HENNA KOIVUSALO, JAMES J. WALTON
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 41 / Issue 5 / May 2021
    Published online by Cambridge University Press:
    20 February 2020, pp. 1431-1463
    Print publication:
    May 2021
    • Article
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  • 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.