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On critical exponents in fixed points of non-erasing morphisms. (English) Zbl 1227.68074
Ibarra, Oscar H. (ed.) et al., Developments in language theory. 10th international conference, DLT 2006, Santa Barbara, CA, USA, June 26–29, 2006. Proceedings. Berlin: Springer (ISBN 3-540-35428-X/pbk). Lecture Notes in Computer Science 4036, 280-291 (2006).
Let ${\Sigma }$ be an alphabet of size $t$, let $f:{{\Sigma }}^{*}\to {{\Sigma }}^{*}$ be a non-erasing morphism, let $w$ be an infinite fixed point of $f$, and let $E\left(w\right)$ be the critical exponent of $w$. We prove that if $E\left(w\right)$ is finite, then for a uniform $f$ it is rational, and for a non-uniform $f$ it lies in the field extension $ℚ\left[{\lambda }_{1},...,{\lambda }_{\ell }\right]$, where ${\lambda }_{1},\cdots ,{\lambda }_{\ell }$ are the eigenvalues of the incidence matrix of $f$. In particular, $E\left(w\right)$ is algebraic of degree at most $t$. Under certain conditions, our proof implies an algorithm for computing $E\left(w\right)$.
##### MSC:
 68Q70 Algebraic theory of languages and automata 68R15 Combinatorics on words
##### Keywords:
critical exponent; circular D0L languages