Hostname: page-component-7dd5485656-pnlb5 Total loading time: 0.001 Render date: 2025-10-30T18:06:03.020Z Has data issue: false hasContentIssue false
  • English
  • Français

On the structure of(−β)-integers

Published online by Cambridge University Press:  07 October 2011

Wolfgang Steiner
Wolfgang Steiner*
Affiliation:
LIAFA, CNRS, Université Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, France.. steiner@liafa.jussieu.fr
Get access

Abstract

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

Keywords

Beta expansionParry numberbeta-integermorphismsubstitution

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

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].
Balková, L., Gazeau, J.-P. and Pelantová, E., Asymptotic behavior of beta-integers. Lett. Math. Phys. 84 (2008) 179–198. Google Scholar
Balková, L., Pelantová, E. and Steiner, W., Sequences with constant number of return words. Monatsh. Math. 155 (2008) 251–263. Google Scholar
Bernat, J., Masáková, Z. and Pelantová, E., On a class of infinite words with affine factor complexity. Theoret. Comput. Sci. 389 (2007) 12–25. Google Scholar
Berthé, V. and Siegel, A., Tilings associated with beta-numeration and substitutions. Integers 5 (2005) 46 (electronic only). Google Scholar
Burdík, Č., Frougny, C., Gazeau, J.P. and Krejcar, R., Beta-integers as natural counting systems for quasicrystals. J. Phys. A 31 (1998) 6449–6472. Google Scholar
Durand, F., A characterization of substitutive sequences using return words. Discrete Math. 179 (1998) 89–101. Google Scholar
Enomoto, F., AH-substitution and Markov partition of a group automorphism on T d. Tokyo J. Math. 31 (2008) 375–398. Google Scholar
Fabre, S., Substitutions et β-systèmes de numération. Theoret. Comput. Sci. 137 (1995) 219–236. Google Scholar
Frougny, C. and Lai, A.C., On negative bases, Proceedings of DLT 09. Lect. Notes Comput. Sci. 5583 (2009) 252–263. Google Scholar
Frougny, C., Masáková, Z. and Pelantová, E., 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. Google Scholar
Gazeau, J.-P. and Verger-Gaugry, J.-L., Geometric study of the beta-integers for a Perron number and mathematical quasicrystals. J. Théor. Nombres Bordeaux 16 (2004) 125–149. Google Scholar
Góra, P., Invariant densities for generalized β-maps. Ergod. Theory Dyn. Syst. 27 (2007) 1583–1598. Google Scholar
Ito, S. and Sadahiro, T., Beta-expansions with negative bases. Integers 9 (2009) 239–259. Google Scholar
C. Kalle and W. Steiner, Beta-expansions, natural extensions and multiple tilings associated with Pisot units. Trans. Am. Math. Soc., to appear.
Klouda, K. and Pelantová, E., Factor complexity of infinite words associated with non-simple Parry numbers. Integers 9 (2009) 281–310. Google Scholar
L. Liao and W. Steiner, Dynamical properties of the negative beta-transformation. To appear in Ergod. Theory Dyn. Syst. arXiv:1101.2366v2.
Masáková, Z. and Pelantová, E., Ito-Sadahiro numbers vs. Parry numbers. Acta Polytech. 51 (2011) 59–64. Google Scholar
Parry, W., On the β-expansions of real numbers. Acta Math. Acad. Sci. Hung. 11 (1960) 401–416. Google Scholar
Rényi, A., Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8 (1957) 477–493. Google Scholar
W. Thurston, Groups, tilings and finite state automata. AMS Colloquium Lectures (1989).