Eigenvalues and Transduction of Morphic Sequences

  • David Sprunger
  • William Tune
  • Jörg Endrullis
  • Lawrence S. Moss
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8633)


We study finite state transduction of automatic and morphic sequences. Dekking [4] proved that morphic sequences are closed under transduction and in particular morphic images. We present a simple proof of this fact, and use the construction in the proof to show that non-erasing transductions preserve a condition called α-substitutivity. Roughly, a sequence is α-substitutive if the sequence can be obtained as the limit of iterating a substitution with dominant eigenvalue α. Our results culminate in the following fact: for multiplicatively independent real numbers α and β, if v is a α-substitutive sequence and w is an β-substitutive sequence, then v and w have no common non-erasing transducts except for the ultimately periodic sequences. We rely on Cobham’s theorem for substitutions, a recent result of Durand [5].


Incidence Matrix Dominant Eigenvalue Input Alphabet Morphic Image Output Alphabet 
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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • David Sprunger
    • 1
  • William Tune
    • 1
  • Jörg Endrullis
    • 1
    • 2
  • Lawrence S. Moss
    • 1
  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA
  2. 2.Department of Computer ScienceVrije Universiteit AmsterdamAmsterdamThe Netherlands

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