Cut and project sets with polytopal window I: complexity

Koivusalo, H. and Walton, J. J. (2021) Cut and project sets with polytopal window I: complexity. Ergodic Theory and Dynamical Systems, 41(5), pp. 1431-1463. (doi: 10.1017/etds.2020.10)

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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Walton, Dr Jamie
• Authors: Koivusalo, H., and Walton, J. J.
• College/School: College of Science and Engineering > School of Mathematics and Statistics
• Journal Name: Ergodic Theory and Dynamical Systems
• Publisher: Cambridge University Press
• ISSN: 0143-3857
• ISSN (Online): 1469-4417
• Published Online: 20 February 2020
• Copyright Holders: Copyright © 2020 The Authors
• First Published: First published in Ergodic Theory and Dynamical Systems 41(5): 1431-1463
• Publisher Policy: Reproduced in accordance with the publisher copyright policy

Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
300655		Applications of space filling curves to substitution tilings	Michael Whittaker	Engineering and Physical Sciences Research Council (EPSRC)	EP/R013691/1	M&S - Mathematics