The unimodularity of the induced toric tilings

V. G. Zhuravlev

Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 1, Tome 469 (2018), pp. 64-95

Résumé

Induced tilings $\mathcal T=\mathcal T|_\mathrm{Kr}$ of the $d$-dimensional torus $\mathbb T^d$, generated by the embedded karyon $\mathrm{Kr}$, are considered. The operations of differentiation are defined $\sigma\colon\mathcal T\to\mathcal T^\sigma$, as a result we get again induced partitions $\mathcal T^\sigma=\mathcal T|_{\mathrm{Kr}^\sigma}$ of the same torus $\mathbb T^d$, generated by the derived karyon $\mathrm{Kr}^\sigma$. In the language of the karyons $\mathrm {Kr}$ the derivations of $\sigma$ reduce to a combination of geometric transformations of the space $\mathbb R^d$. It is proved that if the karyon $\mathrm{Kr}$ is unimodular, then it generates an induced tiling $\mathcal T=\mathcal T|_\mathrm{Kr}$ and the derivative karyon $\mathrm{Kr}^\sigma$ is unimodular again. So there exists the corresponding derivative tiling $\mathcal T^\sigma=\mathcal T|_{\mathrm {Kr}^\sigma}$. Using unimodular karyons one can build an infinite family of induced tilings $\mathcal T=\mathcal T(\alpha,\mathrm{Kr}_*)$ depending on a shift vector $\alpha$ of the torus $\mathbb T^d$ and the initial karyon $\mathrm{Kr}_*$. Two algorithms are presented for constructing such unimodular karyons of $\mathrm{Kr}_*$.

Bibliographie

[1] V. G. Zhuravlev, “Delyaschiesya razbieniya tora i mnozhestva ogranichennogo ostatka”, Zap. nauchn. semin. POMI, 440, 2015, 99–122 | MR

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

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

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

[5] Z. Coelho, A. Lopes, L. F. Da Rocha, “Absolutely Continuous Invariant Measures for a Class of Affine Interval Exchange Maps”, Proccedings of The American Mathematical Society, 123:11 (1995), 3533–3542 | DOI | MR | Zbl

[6] V. G. Zhuravlev, A. V. Shutov, “Derivaties of circle rotations and similarity of orbits”, Max-Planck-Institut für Mathematik Preprint Series, 62 (2004), 1–11 | MR

[7] G. Rauzy, “Ensembles à restes bornés”, Séminaire de théorie des nombres de Bordeaux, 1984, Exposé 24 | MR | Zbl

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

[9] 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 | Zbl

[10] V. G. Zhuravlev, “Mnogomernaya teorema Gekke o raspredelenii drobnykh chastei”, Algebra i analiz, 24:1 (2012), 95–130 | MR | Zbl

[11] S. Grepstad, N. Lev, “Sets of bounded discrepancy for multi-dimensional irrational rotation”, Geometric and Functional Analysis, 25:1 (2014), 87–133 | DOI | MR

[12] V. G. Zhuravlev, “Mnozhestva ogranichennogo ostatka”, Zap. nauchn. semin. POMI, 445, 2016, 93–174 | MR

[13] M. Furukado, Sh. Ito, A. Saito, J. Tamura, Sh. Yasutomi, “A new multidimensional slow continued fraction algorithm and stepped surface”, Experemental Mathematics, 23:4 (2014), 390–410 | DOI | MR | Zbl

[14] V. G. Zhuravlev, “Dvumernye priblizheniya metodom delyaschikhsya toricheskikh razbienii”, Zap. nauchn. semin. POMI, 440, 2015, 81–98 | MR

[15] V. G. Zhuravlev, “Simpleks-modulnyi algoritm razlozheniya algebraicheskikh chisel v mnogomernye tsepnye drobi”, Zap. nauchn. semin. POMI, 449, 2016, 168–195 | MR

[16] V. G. Zhuravlev, “Periodicheskie yadernye razlozheniya kubicheskikh irratsionalnostei v tsepnye drobi”, Sovr. probl. matem., 23, MIAN, 2016, 43–68 | DOI

[17] M. Morse, C. A. Hedlund, “Symbolic Dynamicsyu II: Sturmian trajectories”, Amer. J. Math., 62 (1940), 1–42 | DOI | MR | Zbl

[18] V. G. Zhuravlev, “Moduli toricheskikh razbienii na mnozhestva ogranichennogo ostatka i sbalansirovannye slova”, Algebra i analiz, 24:4 (2012), 97–136 | MR | Zbl

[19] V. G. Zhuravlev, A. V. Maleev, “Posloinyi rost kvaziperiodicheskogo razbieniya Rozi”, Kristallografiya, 52:2 (2007), 204–210

[20] A. V. Shutov, A. V. Maleev, V. G. Zhuravlev, “Complex quasiperiodic self-similar tilings: their parameterization, boundaries, complexity, growth and symmetry”, Acta Crystallogr. A, 66 (2010), 427–437 | DOI | MR

[21] V. G. Zhuravlev, “On additive property of a complexity function related to Rauzy tiling”, Anal. Probab. Methods Number Theory, eds. E. Manstavicius et al., TEV, Vilnius, 2007, 240–254 | MR | Zbl

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

[23] G. F. Voronoi, Sobranie sochinenii, v. 2, Kiev, 1952