Let $\mathcal{A}$ be some alphabet of distinct symbols, say $\mathcal{A} = \{a, f, l\}$. Let $\mathcal{A}^*$ be the set of all finite words over $\mathcal{A}$. An overlap is any word of the form $xyxyx$, where $x$ and $y$ are subwords and $x$ is non-empty. A word is periodic (or a proper power) if $w = x^n$ for some word $x$ and some $n>1$.
For example, $alfalfa$ is an overlap, with $x = a$ and $y = lf$. The word $alfalfalf$ is periodic.
Note that $alfalfa$ has a cyclic rotation $lfalfaa$ which has no overlap prefix.
If $w \in \mathcal{A}^*$ is aperiodic, is there a cyclic rotation of $w$ that has no overlap prefix?
An aperiodic word need not have a cyclic rotation with no periodic prefix. For example, every cyclic rotation of $w = (faa)^3 fa$ has a periodic prefix.