On the structure of (-ε)-integers
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 46 (2012) no. 1, pp. 181-200.

The (-β)-integers are natural generalisations of the β-integers, and thus of the integers, for negative real bases. When β is the analogue of a Parry number, we describe the structure of the set of (-β)-integers by a fixed point of an anti-morphism.

DOI: 10.1051/ita/2011115
Keywords: beta expansion, Parry number, beta-integer, morphism, substitution
@article{ITA_2012__46_1_181_0,
     author = {Steiner, Wolfgang},
     title = {On the structure of $(-\varepsilon )$-integers},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {181--200},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {1},
     year = {2012},
     doi = {10.1051/ita/2011115},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita/2011115/}
}
TY  - JOUR
AU  - Steiner, Wolfgang
TI  - On the structure of $(-\varepsilon )$-integers
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2012
DA  - 2012///
SP  - 181
EP  - 200
VL  - 46
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ita/2011115/
UR  - https://doi.org/10.1051/ita/2011115
DO  - 10.1051/ita/2011115
LA  - en
ID  - ITA_2012__46_1_181_0
ER  - 
%0 Journal Article
%A Steiner, Wolfgang
%T On the structure of $(-\varepsilon )$-integers
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2012
%P 181-200
%V 46
%N 1
%I EDP-Sciences
%U https://doi.org/10.1051/ita/2011115
%R 10.1051/ita/2011115
%G en
%F ITA_2012__46_1_181_0
Steiner, Wolfgang. On the structure of $(-\varepsilon )$-integers. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 46 (2012) no. 1, pp. 181-200. doi : 10.1051/ita/2011115. http://www.numdam.org/articles/10.1051/ita/2011115/

[1] P. Ambrož, D. Dombek, Z. Masáková and E. Pelantová, Numbers with integer expansion in the numeration system with negative base. arXiv:0912.4597v3 [math.NT]. | MR | Zbl

[2] L. Balková, J.-P. Gazeau and E. Pelantová, Asymptotic behavior of beta-integers. Lett. Math. Phys. 84 (2008) 179-198. | MR | Zbl

[3] L. Balková, E. Pelantová and W. Steiner, Sequences with constant number of return words. Monatsh. Math. 155 (2008) 251-263. | MR | Zbl

[4] J. Bernat, Z. Masáková and E. Pelantová, On a class of infinite words with affine factor complexity. Theoret. Comput. Sci. 389 (2007) 12-25. | MR | Zbl

[5] V. Berthé and A. Siegel, Tilings associated with beta-numeration and substitutions. Integers 5 (2005) 46 (electronic only). | MR | Zbl

[6] Č. Burdík, C. Frougny, J.P. Gazeau and R. Krejcar, Beta-integers as natural counting systems for quasicrystals. J. Phys. A 31 (1998) 6449-6472. | MR | Zbl

[7] F. Durand, A characterization of substitutive sequences using return words. Discrete Math. 179 (1998) 89-101. | MR | Zbl

[8] F. Enomoto, AH-substitution and Markov partition of a group automorphism on Td. Tokyo J. Math. 31 (2008) 375-398. | MR | Zbl

[9] S. Fabre, Substitutions et β-systèmes de numération. Theoret. Comput. Sci. 137 (1995) 219-236. | MR | Zbl

[10] C. Frougny and A.C. Lai, On negative bases, Proceedings of DLT 09. Lect. Notes Comput. Sci. 5583 (2009) 252-263. | MR | Zbl

[11] C. Frougny, Z. Masáková and E. Pelantová, Complexity of infinite words associated with beta-expansions. RAIRO-Theor. Inf. Appl. 38 (2004) 163-185; Corrigendum: RAIRO-Theor. Inf. Appl. 38 (2004) 269-271. | Numdam | MR | Zbl

[12] J.-P. Gazeau and J.-L. Verger-Gaugry, Geometric study of the beta-integers for a Perron number and mathematical quasicrystals. J. Théor. Nombres Bordeaux 16 (2004) 125-149. | Numdam | MR | Zbl

[13] P. Góra, Invariant densities for generalized β-maps. Ergod. Theory Dyn. Syst. 27 (2007) 1583-1598. | Zbl

[14] S. Ito and T. Sadahiro, Beta-expansions with negative bases. Integers 9 (2009) 239-259. | MR | Zbl

[15] C. Kalle and W. Steiner, Beta-expansions, natural extensions and multiple tilings associated with Pisot units. Trans. Am. Math. Soc., to appear. | MR | Zbl

[16] K. Klouda and E. Pelantová, Factor complexity of infinite words associated with non-simple Parry numbers. Integers 9 (2009) 281-310. | MR | Zbl

[17] L. Liao and W. Steiner, Dynamical properties of the negative beta-transformation. To appear in Ergod. Theory Dyn. Syst. arXiv:1101.2366v2. | MR | Zbl

[18] Z. Masáková and E. Pelantová, Ito-Sadahiro numbers vs. Parry numbers. Acta Polytech. 51 (2011) 59-64.

[19] W. Parry, On the β-expansions of real numbers. Acta Math. Acad. Sci. Hung. 11 (1960) 401-416. | MR | Zbl

[20] A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8 (1957) 477-493. | MR | Zbl

[21] W. Thurston, Groups, tilings and finite state automata. AMS Colloquium Lectures (1989).

Cited by Sources: