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We prove that, for any natural number $$n\ge 1$$ n ≥ 1 , we can find a finite alphabet $$\Sigma $$ Σ and a finitary language L over $$\Sigma $$ Σ accepted by a one-counter automaton, such that the $$\omega $$ ω -power $$\begin{aligned} L^\infty :=\{ w_0w_1\ldots \in \Sigma ^\omega \mid \forall i\in \omega ~~w_i\in L\} \end{aligned}$$ L ∞ : = { w 0 w 1 … ∈ Σ ω ∣ ∀ i ∈ ω w i ∈ L } is $${\varvec{\Pi }}^0_n$$ Π n 0 -complete. We prove a similar result for the class $${\varvec{\Sigma }}^0_n$$ Σ n 0 .

Some complete $$\omega $$-powers of a one-counter language, for any Borel class of finite rank