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The zeta function of a finite automaton A is exp{ ∞ n=1 a n z n n }, where a n is the number of bi-infinite paths in A labelled by a bi-infinite word of period n. It reflects the properties of A: aperiodicity, nil-simplicity, existence of a zero. The results are applied to codes.

We determine the left eigenvector of a stochastic matrix associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is , where is the M 1

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