A classification of one dimensional almost periodic tilings arising from the projection method
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- by James A. Mingo PDF
- Trans. Amer. Math. Soc. 352 (2000), 5263-5277
Abstract:
For each irrational number $\alpha$, with continued fraction expansion $[0; a_1, a_2,a_3, \dots ]$, we classify, up to translation, the one dimensional almost periodic tilings which can be constructed by the projection method starting with a line of slope $\alpha$. The invariant is a sequence of integers in the space $X_\alpha = \{(x_i)_{i=1}^\infty \mid x_i \in \{0,1,2, \dots ,a_i\}$ and $x_{i+1} = 0$ whenever $x_i = a_i\}$ modulo the equivalence relation generated by tail equivalence and $(a_1, 0, a_3, 0, \dots ) \sim (0, a_2, 0, a_4, \dots ) \sim (a_1 -1, a_2 - 1, a_3 - 1, \dots )$. Each tile in a tiling $\textsf {T}$, of slope $\alpha$, is coded by an integer $0 \leq x \leq [\alpha ]$. Using a composition operation, we produce a sequence of tilings $\textsf {T}_1 = \textsf {T}{}, \textsf {T}_2, \textsf {T}_3, \dots$. Each tile in $\textsf {T}_i$ gets absorbed into a tile in $\textsf {T}_{i+1}$. A choice of a starting tile in $\textsf {T}_1$ will thus produce a sequence in $X_\alpha$. This is the invariant.References
- Jared E. Anderson and Ian F. Putnam, Topological invariants for substitution tilings and their associated $C^*$-algebras, Ergodic Theory Dynam. Systems 18 (1998), no. 3, 509–537. MR 1631708, DOI 10.1017/S0143385798100457
- Jean Bernoulli (III), (1744-1807), Recueil pour les Astromomes, tome 1, Berlin, chez l’auteur, (1771).
- Tom C. Brown, Descriptions of the characteristic sequence of an irrational, Canad. Math. Bull. 36 (1993), no. 1, 15–21. MR 1205889, DOI 10.4153/CMB-1993-003-6
- N. G. de Bruijn, Sequences of zeros and ones generated by special production rules, Nederl. Akad. Wetensch. Indag. Math. 43 (1981), no. 1, 27–37. MR 609464, DOI 10.1016/1385-7258(81)90015-9
- N. G. de Bruijn, Updown generation of Beatty sequences, Nederl. Akad. Wetensch. Indag. Math. 51 (1989), no. 4, 385–407. MR 1041493, DOI 10.1016/1385-7258(89)90003-6
- E. B. Christoffel, Observatio Arithmetica, Annali di Mathematica, (2) 6 (1875) 148-152.
- Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
- Branko Grünbaum and G. C. Shephard, Tilings and patterns, A Series of Books in the Mathematical Sciences, W. H. Freeman and Company, New York, 1989. An introduction. MR 992195
- W. F. Lunnon and P. A. B. Pleasants, Characterization of two-distance sequences, J. Austral. Math. Soc. Ser. A 53 (1992), no. 2, 198–218. MR 1175712, DOI 10.1017/S1446788700035795
- James A. Mingo, $C^*$-algebras associated with one-dimensional almost periodic tilings, Comm. Math. Phys. 183 (1997), no. 2, 307–337. MR 1461961, DOI 10.1007/BF02506409
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416 (English, with Russian summary). MR 142719, DOI 10.1007/BF02020954
- A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493. MR 97374, DOI 10.1007/BF02020331
- E. Arthur Robinson Jr., The dynamical properties of Penrose tilings, Trans. Amer. Math. Soc. 348 (1996), no. 11, 4447–4464. MR 1355301, DOI 10.1090/S0002-9947-96-01640-6
- Marjorie Senechal, Quasicrystals and geometry, Cambridge University Press, Cambridge, 1995. MR 1340198
- Caroline Series, The geometry of Markoff numbers, Math. Intelligencer 7 (1985), no. 3, 20–29. MR 795536, DOI 10.1007/BF03025802
- H. J. S. Smith, A Note on Continued Fractions, Messenger of Math., (2) 6 (1876) 1-14.
Additional Information
- James A. Mingo
- Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada
- Email: mingoj@mast.queensu.ca
- Received by editor(s): August 4, 1998
- Received by editor(s) in revised form: May 1, 1999
- Published electronically: July 18, 2000
- Additional Notes: Research supported by the Natural Sciences and Engineering Research Council of Canada and The Fields Institute for Research in the Mathematical Sciences
- © Copyright 2000 by the author
- Journal: Trans. Amer. Math. Soc. 352 (2000), 5263-5277
- MSC (1991): Primary 05B45, 52C22, 46L89
- DOI: https://doi.org/10.1090/S0002-9947-00-02620-9
- MathSciNet review: 1709776