  We study a new notion of cyclic avoidance of abelian powers. A finite word
$w$ avoids abelian $N$-powers cyclically if for each abelian $N$-power of
period $m$ occurring in the infinite word $w^\omega$, we have $m \geq |w|$. Let
$\mathcal{A}(k)$ be the least integer $N$ such that for all $n$ there exists a
word of length $n$ over a $k$-letter alphabet that avoids abelian $N$-powers
cyclically. Let $\mathcal{A}_\infty(k)$ be the least integer $N$ such that
there exist arbitrarily long words over a $k$-letter alphabet that avoid
abelian $N$-powers cyclically.
  We prove that $5 \leq \mathcal{A}(2) \leq 8$, $3 \leq \mathcal{A}(3) \leq 4$,
$2 \leq \mathcal{A}(4) \leq 3$, and $\mathcal{A}(k) = 2$ for $k \geq 5$.
Moreover, we show that $\mathcal{A}_\infty(2) = 4$, $\mathcal{A}_\infty(3) =
3$, and $\mathcal{A}_\infty(4) = 2$.

Formal Languages and Automata Theory