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A graph approach to computing nondeterminacy in substitutional dynamical systems

Published online by Cambridge University Press:  25 September 2007

Toke M. Carlsen  and
Søren Eilers
Toke M. Carlsen
Affiliation:
University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark; toke@math.ku.dk School of Mathematical & Physical Sciences, Mathemathics Building - V123, University of Newcastle, University Drive, Callaghan NSW 2308, Australia.
Søren Eilers
Affiliation:
University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark; eilers@math.ku.dk
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Abstract

We present an algorithm which for any aperiodic and primitive substitution outputs a finite representation of each special word in the shift space associated to that substitution, and determines when such representations are equivalent under orbit and shift tail equivalence. The algorithm has been implemented and applied in the study of certain new invariants for flow equivalence of substitutional dynamical systems.

Keywords

Sustitutionshift spacesspecial elementsorbit equivalenceshift tail equivalence
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 41 , Issue 3 , July 2007 , pp. 285 - 306
Copyright
© EDP Sciences, 2007

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References

Barge, M. and Diamond, B., A complete invariant for the topology of one-dimensional substitution tiling spaces. Ergodic Theory Dynam. Systems 21 (2001) 1333–1358. CrossRef
Barge, M., Diamond, B. and Holton, C., Asymptotic orbits of primitive substitutions. Theoret. Comput. Sci. 301 (2003) 439–450. CrossRef
Boyle, M. and Lind, D., Exponential subdynamics. Trans. Amer. Math. Soc. 349 (1997) 55–102. CrossRef
T.M. Carlsen and S. Eilers, Java applet, www.math.ku.dk/ẽilers/papers/cei (2002).
Carlsen, T.M. and Eilers, S., Augmenting dimension group invariants for substitution dynamics. Ergodic Theory Dynam. Systems 24 (2004) 1015–1039. CrossRef
Carlsen, T.M. and Eilers, S., Matsumoto K-groups associated to certain shift spaces. Doc. Math. 9 (2004) 639–671.
Carlsen, T.M. and Eilers, S., Ordered K-groups associated to substitutional dynamics. J. Funct. Anal. 238 (2006) 99–117. CrossRef
Durand, F., Host, B., and Skau, C., Substitutional dynamical systems, Bratteli diagrams and dimension groups. Ergodic Theory Dynam. Systems 19 (1999) 953–993. CrossRef
N.P. Fogg, Substitutions in dynamics, arithmetics and combinatorics. Lect. Notes Math. 1794 Springer-Verlag, Heidelberg (2002).
Holton, C. and Zamboni, L.Q., Directed graphs and substitutions. Theory Comput. Syst. 34 (2001) 545–564. CrossRef
B.P. Kitchens, Symbolic dynamics; One-sided, two-sided and countable state Markov shifts. Springer-Verlag, Berlin (1998).
D. Lind and B. Marcus, An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge (1995).
Matsumoto, K., Bowen-Franks groups as an invariant for flow equivalence of subshifts. Ergodic Theory Dynam. Systems 21 (2001) 1831–1842.
Mossé, B., Puissances de mots et reconnaissabilité des points fixes d'une substitution. Theoret. Comput. Sci. 99 (1992) 327–334. CrossRef
Pansiot, J.-J., Decidability of periodicity for infinite words. RAIRO-Theor. Inf. Appl. 20 (1986) 43–46. CrossRef
Parry, B. and Sullivan, D., A topological invariant of flows on 1-dimensional spaces. Topology 14 (1975) 297–299. CrossRef
M. Queffélec, Substitution dynamical systems—spectral analysis. Springer-Verlag, Berlin (1987).
G. Rozenberg and A. Salomaa, The mathematical theory of L systems. Academic Press Inc. Harcourt Brace Jovanovich Publishers, New York (1980).
Wielandt, H., Unzerlegbare, nicht negative Matrizen. Math. Z. 52 (1950) 642–648. CrossRef