Inflation and deflation of the karyon tilings

V. G. Zhuravlev

Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 6, Tome 523 (2023), pp. 53-82

Résumé

The substitution transformations of inflation and deflation are defined for the karyon tilings $\mathcal{T}(v)$ of multidimensional tori $\mathbb{T}^d$. Such tilings $\mathcal{T}(v)$ consist of parallelepipeds and are generated by its karyons. Stars $v$, sets of $d+1$ vectors in the space $\mathbb{R}^d$, are frames of the karyons. The interest in karyon tilings is due to their connections with multidimensional continued fractions.

Bibliographie

[1] N. G. de Bruijn, “Algebraic theory of Penrose's non-periodic tilings of the plane I, II”, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen Series A, 84:1 (1981), 39–66 | MR

[2] B. Grünbaum, G. C. Shephard, Tilings and Patterns, W. H. Freeman, San Francisco, 1987 | MR | Zbl

[3] M. Gardner, Penrose Tiles to Trapdoor Ciphers and the return of Dr. Matrix, W. H. Freeman and Co., New York, 1989 | MR

[4] P. Arnoux, S. Ito, “Pisot Substitutions and Rauzy fractals”, Bulletin of the Belgian Mathematical Society, 8:2 (2001), 1–27 | MR

[5] S. Ito, “Diophantine approximations, substitutions, and fractals”, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, 1794, ed. N.P. Fogg, Springer, Berlin–Heidelberg, 2002 | MR

[6] V. G. Zhuravlev, A. V. Maleev, “Simmetriya podobiya dvumernogo kvaziperiodicheskogo razbieniya Rozi”, Kristallografiya, 54 (2009), 400–409

[7] S. Akiyama, J-Y. Lee, “Determining quasicrystal structures on substitution tilings”, Philosophical Magazine A, 91:19 (2011), 2709–2717 | DOI

[8] V. G. Zhuravlev, “Differentsirovanie indutsirovannykh razbienii tora i mnogomernye priblizheniya algebraicheskikh chisel”, Zap. nauchn. semin. POMI, 445, 2016, 33–92

[9] G. Rauzy, “Nombres algèbriques et substitutions”, Bull. Soc. Math. France, 110 (1982), 147–178 | DOI | MR | Zbl

[10] V. G. Zhuravlev, “Razbieniya Rozi i mnozhestva ogranichennogo ostatka na tore”, Zap. nauchn. semin. POMI, 322, 2005, 83–106 | Zbl

[11] A. V. Shutov, A. V. Maleev, V. G. Zhuravlev, “Complex quasiperiodic self-similar tilings: their parameterization, boundaries, complexity, growth and symmetry”, Acta Crystallogr., A66 (2010), 427–437 | DOI | MR

[12] V. G. Zhuravlev, “Odnomernye razbieniya Fibonachchi”, Izv. RAN, ser. matem., 71:2 (2007), 89–122 | DOI | MR | Zbl

[13] V. G. Zhuravlev, “Universalnye yadernye razbieniya”, Zap. nauchn. semin. POMI, 490, 2020, 49–93

[14] V. G. Zhuravlev, “Lokalnyi algoritm postroeniya proizvodnykh razbienii dvumernogo tora”, Zap. nauchn. semin. POMI, 479, 2019, 85–120

[15] V. G. Zhuravlev, “Simpleks-yadernyi algoritm razlozheniya v mnogomernye tsepnye drobi”, Sovremennye problemy matem., 299, MIAN, 2017, 283–303

[16] S. Ito, M. Ohtsuki, “Parallelogram tilings and Jacobi-Perron algorithm”, Tokyo J. Math., 17:1 (1994), 33–58 | DOI | MR | Zbl

[17] P. Arnoux, V. Berthé, H. Ei, Sh. Ito, “Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions”, Maison de l'Informatique et des Mathématiques Discrètes (MIMD) (Paris, 2001), 59–78 | MR | Zbl

[18] V. G. Zhuravlev, Yadernye tsepnye drobi, VlGU, Vladimir, 2019

[19] V. G. Zhuravlev, “Perekladyvayuschiesya toricheskie razvertki i mnozhestva ogranichennogo ostatka”, Zap. nauchn. semin. POMI, 392, 2011, 95–145

[20] V. G. Zhuravlev, “Mnogogranniki ogranichennogo ostatka”, Matematika i informatika, K 75-letiyu so dnya rozhdeniya Anatoliya Alekseevicha Karatsuby, v. 1, Sovr. probl. matem., 16, MIAN, M., 2012, 82–102 | DOI

[21] E. S. Fedorov, Nachala ucheniya o figurakh, M., 1953 | MR

[22] G. F. Voronoi, Sobranie sochinenii, v. 2, Kiev, 1952 | MR