Geodesic
Self-similarity and substitutions of the karyon tilings
Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 6, Tome 523 (2023), pp. 83-120 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Self-similar karyon partitions $\mathcal{T}(\mathbf{m},v)$ with parameters the weight vector $\mathbf{m}$ and the star $v$ are considered. The star $v$ defines the geometry of the parallelepipeds of which the tiling consists of and the weight vector $\mathbf{m}$ sets local rules and periodicity of $\mathcal{T}(\mathbf{m},v)$. A deflation $\bigtriangleup:\mathcal{T}(\mathbf{m},v) \longrightarrow \mathcal{T}^{\bigtriangleup}(\mathbf{m},v)$ is being built, where $\mathcal{T}^{\bigtriangleup}(\mathbf{m},v)=A\mathcal{T}(\mathbf{m},v)$, and $A$ is an affine mapping of the space $\mathbb{R}^{d}$. Deflation replaces the basic polyhedra forming the tiling $\mathcal{T}(\mathbf{m},v)$ by smaller polyhedra. This is the main idea of multidimensional approximations by continued fractions. 
@article{ZNSL_2023_523_a5,
     author = {V. G. Zhuravlev},
     title = {Self-similarity and substitutions of the karyon tilings},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {83--120},
     year = {2023},
     volume = {523},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2023_523_a5/}
}
TY  - JOUR
AU  - V. G. Zhuravlev
TI  - Self-similarity and substitutions of the karyon tilings
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2023
SP  - 83
EP  - 120
VL  - 523
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2023_523_a5/
LA  - ru
ID  - ZNSL_2023_523_a5
ER  - 
%0 Journal Article
%A V. G. Zhuravlev
%T Self-similarity and substitutions of the karyon tilings
%J Zapiski Nauchnykh Seminarov POMI
%D 2023
%P 83-120
%V 523
%U http://geodesic.mathdoc.fr/item/ZNSL_2023_523_a5/
%G ru
%F ZNSL_2023_523_a5
V. G. Zhuravlev. Self-similarity and substitutions of the karyon tilings. Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 6, Tome 523 (2023), pp. 83-120. http://geodesic.mathdoc.fr/item/ZNSL_2023_523_a5/

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

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

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

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

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

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

[7] V. G. Zhuravlev, “Differentsirovanie yadernykh razbienii”, Zap. nauchn. semin. POMI, 511, 2022, 28–53

[8] V. G. Zhuravlev, “Yadernye razbieniya i mnogomernye tsepnye drobi”, Vtoraya konferentsiya matematicheskikh tsentrov (07.11–11.11, Moskva, 2022), 90–92 | Zbl

[9] V. G. Zhuravlev, “Inflyatsiya i deflyatsiya yadernykh razbienii”, Zap. nauch. semin. POMI, 523, 2023, 53–82

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

[11] P. Arnoux, V. Berthé, S. Ito, “Discrete planes, $\mathbb{Z}^2$-actions, Jacobi-Perron algorithm and substitutions”, Ann. Inst. Fourier (Grenoble), 52:2 (2002), 305–349 | DOI | MR | Zbl

[12] V. Berthé, L. Vuillon, “Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences”, Discrete Math., 223 (2000), 27–53 | DOI | MR | Zbl

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

[14] 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

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

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

[17] 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