Cut and project sets with polytopal window II: linear repetitivity
HTML articles powered by AMS MathViewer
- by Henna Koivusalo and James J. Walton PDF
- Trans. Amer. Math. Soc. 375 (2022), 5097-5149 Request permission
Abstract:
In this paper we give a complete characterisation of linear repetitivity for cut and project schemes with convex polytopal windows satisfying a weak homogeneity condition. This answers a question of Lagarias and Pleasants from the 90s for a natural class of cut and project schemes which is large enough to cover almost all such polytopal schemes which are of interest in the literature. We show that a cut and project scheme in this class has linear repetitivity exactly when it has the lowest possible patch complexity and satisfies a Diophantine condition. Finding the correct Diophantine condition is a major part of the work. To this end we develop a theory, initiated by Forrest, Hunton and Kellendonk, of decomposing polytopal cut and project schemes to factors. We also demonstrate our main theorem on a wide variety of examples, covering all classical examples of canonical cut and project schemes, such as Penrose and Ammann–Beenker tilings.References
- S. Akiyama, M. Barge, V. Berthé, J.-Y. Lee, and A. Siegel, On the Pisot substitution conjecture, Mathematics of aperiodic order, Progr. Math., vol. 309, Birkhäuser/Springer, Basel, 2015, pp. 33–72. MR 3381478, DOI 10.1007/978-3-0348-0903-0_{2}
- Marcy Barge, The Pisot conjecture for $\beta$-substitutions, Ergodic Theory Dynam. Systems 38 (2018), no. 2, 444–472. MR 3774828, DOI 10.1017/etds.2016.44
- Adnene Besbes, Michael Boshernitzan, and Daniel Lenz, Delone sets with finite local complexity: linear repetitivity versus positivity of weights, Discrete Comput. Geom. 49 (2013), no. 2, 335–347. MR 3017915, DOI 10.1007/s00454-012-9455-z
- Marcy Barge and Beverly Diamond, Coincidence for substitutions of Pisot type, Bull. Soc. Math. France 130 (2002), no. 4, 619–626 (English, with English and French summaries). MR 1947456, DOI 10.24033/bsmf.2433
- F. P. M. Beenker, Algebraic theory of non-periodic tilings of the plane by two simple building blocks: a square and a rhombus, EUT Report, WSK, Dept. of Mathematics and Computing Science, Eindhoven University of Technology, 1982 (English).
- Valérie Berthé and Thomas Fernique, Brun expansions of stepped surfaces, Discrete Math. 311 (2011), no. 7, 521–543. MR 2765621, DOI 10.1016/j.disc.2010.12.007
- Nicolas Bédaride and Thomas Fernique, Weak local rules for planar octagonal tilings, Israel J. Math. 222 (2017), no. 1, 63–89. MR 3736499, DOI 10.1007/s11856-017-1582-z
- Riccardo Benedetti and Jean-Marc Gambaudo, On the dynamics of $\Bbb G$-solenoids. Applications to Delone sets, Ergodic Theory Dynam. Systems 23 (2003), no. 3, 673–691. MR 1992658, DOI 10.1017/S0143385702001578
- Michael Baake and Uwe Grimm, Aperiodic order. Vol. 1, Encyclopedia of Mathematics and its Applications, vol. 149, Cambridge University Press, Cambridge, 2013. A mathematical invitation; With a foreword by Roger Penrose. MR 3136260, DOI 10.1017/CBO9781139025256
- M. Baake, P. Kramer, M. Schlottmann, and D. Zeidler, Planar patterns with fivefold symmetry as sections of periodic structures in $4$-space, Internat. J. Modern Phys. B 4 (1990), no. 15-16, 2217–2268. MR 1086074, DOI 10.1142/S0217979290001054
- Valérie Berthé, Milton Minervino, Wolfgang Steiner, and Jörg Thuswaldner, The $S$-adic Pisot conjecture on two letters, Topology Appl. 205 (2016), 47–57. MR 3493306, DOI 10.1016/j.topol.2016.01.019
- J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957. MR 0087708
- Alan Forrest, John Hunton, and Johannes Kellendonk, Topological invariants for projection method patterns, Mem. Amer. Math. Soc. 159 (2002), no. 758, x+120. MR 1922206, DOI 10.1090/memo/0758
- Branko GrĂĽnbaum and G. C. Shephard, Tilings and patterns, W. H. Freeman and Company, New York, 1987. MR 857454
- Alan Haynes, Henna Koivusalo, and James Walton, A characterization of linearly repetitive cut and project sets, Nonlinearity 31 (2018), no. 2, 515–539. MR 3755878, DOI 10.1088/1361-6544/aa9528
- Michael Hollander and Boris Solomyak, Two-symbol Pisot substitutions have pure discrete spectrum, Ergodic Theory Dynam. Systems 23 (2003), no. 2, 533–540. MR 1972237, DOI 10.1017/S0143385702001384
- Antoine Julien and Jean Savinien, Transverse Laplacians for substitution tilings, Comm. Math. Phys. 301 (2011), no. 2, 285–318. MR 2764989, DOI 10.1007/s00220-010-1150-4
- Antoine Julien, Complexity and cohomology for cut-and-projection tilings, Ergodic Theory Dynam. Systems 30 (2010), no. 2, 489–523. MR 2599890, DOI 10.1017/S0143385709000194
- Dong Han Kim, The shrinking target property of irrational rotations, Nonlinearity 20 (2007), no. 7, 1637–1643. MR 2335077, DOI 10.1088/0951-7715/20/7/006
- Dmitry Kleinbock, Badly approximable systems of affine forms, J. Number Theory 79 (1999), no. 1, 83–102. MR 1724255, DOI 10.1006/jnth.1999.2419
- Henna Koivusalo and James J. Walton, Cut and project sets with non-convex and disconnected polygonal windows, In preparation.
- Henna Koivusalo and James J. Walton, Cut and project sets with polytopal window I: complexity, Ergodic Theory Dynam. Systems (2020), 1–33.
- J.-Y. Lee, R. V. Moody, and B. Solomyak, Pure point dynamical and diffraction spectra, Ann. Henri Poincaré 3 (2002), no. 5, 1003–1018. MR 1937612, DOI 10.1007/s00023-002-8646-1
- Jeffrey C. Lagarias and Peter A. B. Pleasants, Repetitive Delone sets and quasicrystals, Ergodic Theory Dynam. Systems 23 (2003), no. 3, 831–867. MR 1992666, DOI 10.1017/S0143385702001566
- Marston Morse and Gustav A. Hedlund, Symbolic Dynamics, Amer. J. Math. 60 (1938), no. 4, 815–866. MR 1507944, DOI 10.2307/2371264
- Marston Morse and Gustav A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62 (1940), 1–42. MR 745, DOI 10.2307/2371431
- Farhad A. Namin and Douglas H. Werner, An exact method to determine the photonic resonances of quasicrystals based on discrete Fourier harmonics of higher-dimensional atomic surfaces, Crystals 6 (2016), no. 93.
- R. Penrose, Pentaplexity: a class of nonperiodic tilings of the plane, Math. Intelligencer 2 (1979/80), no. 1, 32–37. MR 558670, DOI 10.1007/BF03024384
- D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984), 1951–1953.
- Martin Schlottmann, Generalized model sets and dynamical systems, Directions in mathematical quasicrystals, CRM Monogr. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2000, pp. 143–159. MR 1798991
Additional Information
- Henna Koivusalo
- Affiliation: School of Mathematics, Fry Building, Woodland Road, Bristol BS8 1UG, United Kingdom
- MR Author ID: 1062599
- Email: henna.koivusalo@bristol.ac.uk
- James J. Walton
- Affiliation: School of Mathematical Sciences, Mathematical Sciences Building, University Park, Nottingham NG7 2RD, United Kingdom
- MR Author ID: 1162597
- ORCID: 0000-0003-3804-444X
- Email: James.Walton@nottingham.ac.uk
- Received by editor(s): May 18, 2021
- Received by editor(s) in revised form: December 23, 2021, and January 3, 2022
- Published electronically: May 4, 2022
- Additional Notes: We gratefully acknowledge the support of the London Mathematical Society, Scheme 2. The research of the second author was partially supported by EPSRC grant EP/R013691/1.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 5097-5149
- MSC (2020): Primary 52C23; Secondary 52C45
- DOI: https://doi.org/10.1090/tran/8633
- MathSciNet review: 4439500