On critical exponents in fixed points of non-erasing morphisms

Let @S be an alphabet of size t, let f:@S^*->@S^* 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 Q[r,@l"1,...,@l"@?], where r,@l"1,...,@l"@? 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).

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