Avoiding abelian powers cyclically

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 $m\ge \mathrm{\mid }w\mathrm{\mid }$. 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 ${\mathcal{A}}_{\mathrm{\infty }}(k)$ 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 $5\le \mathcal{A}(2)\le 8$, $3\le \mathcal{A}(3)\le 4$, $2\le \mathcal{A}(4)\le 3$, and $\mathcal{A}(k)=2$ for $k\ge 5$. Moreover, we show that ${\mathcal{A}}_{\mathrm{\infty }}(2)=4$, $$\mathcal{A}$\infty(3) = 3$,and$$\mathcal{A}$\infty(4) = 2$.