Rocky Mountain Journal of Mathematics

A short proof of a theorem of Cobham on substitutions

Ethan M. Coven, Andrew Dykstra, and Michelle Lemasurier

Full-text: Open access

Abstract

This paper is concerned with the lengths of constant length substitutions that generate topologically conjugate systems. We show that if the systems are infinite, then these lengths must be powers of the same integer. This result is a dynamical formulation of a special case of a 1969 theoretical computer science result of Cobham [{\bf1}]. Our proof is rather simple.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 1 (2014), 19-22.

Dates
First available in Project Euclid: 2 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1401740488

Digital Object Identifier
doi:10.1216/RMJ-2014-44-1-19

Mathematical Reviews number (MathSciNet)
MR3216006

Zentralblatt MATH identifier
1298.37007

Citation

Coven, Ethan M.; Dykstra, Andrew; Lemasurier, Michelle. A short proof of a theorem of Cobham on substitutions. Rocky Mountain J. Math. 44 (2014), no. 1, 19--22. doi:10.1216/RMJ-2014-44-1-19. https://projecteuclid.org/euclid.rmjm/1401740488


Export citation

References

  • A. Cobham, On the base-dependence of sets of numbers recognizable by finite automata, Math. Syst. Theor. 3 (1969), 186-192.
  • E.M. Coven, M. Keane and M. LeMasurier, A characterization of the Morse minimal set up to topological conjugacy, Ergod. Theor. Dynam. Syst. 28 (2008), 1443-1451.
  • F.M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrschein. Verw. Geb. 41 (1977/78), 221-239.
  • F. Durand, Cobham-Semenov theorem and $\n^d$-subshifts, Theor. Comp. Sci. 391 (2008), 20-38.
  • S. Eilenberg, Automata, languages, and machines, Pure Appl. Math. 58, Academic Press, New York, 1974. \noindentstyle