On critical exponents in fixed points of non-erasing morphisms

  • Authors:

  • Dalia Krieger

  • Affiliations:

  • School of Computer Science, University of Waterloo, Waterloo, ON, Canada

  • Venue:
  • DLT'06 Proceedings of the 10th international conference on Developments in Language Theory
  • Year:
  • 2006

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Abstract

Let Σ be an alphabet of size t, let f:Σ*→Σ* be a non-erasing morphism, let w be an infinite fixed point of f, and let E(w) be the critical exponent of w. We prove that if E(w) is finite, then for a uniform f it is rational, and for a non-uniform f it lies in the field extension ${{\mathbb Q}[{\lambda_1},\ldots,{\lambda_\ell}]}$, where λ1,...,λℓ are the eigenvalues of the incidence matrix of f. In particular, E(w) is algebraic of degree at most t. Under certain conditions, our proof implies an algorithm for computing E(w). MSC: 68R15