A function computed by a Mealy machine is aperiodic if and only if its minimal Mealy machine satisfies the stabilisation condition (): for every state transformation $\delta : Q \to Q$ that arises from some input string, the sequence $\delta^1, \delta^2, \ldots$ eventually stabilises on a single state transformation (Lemma A.2.11 of Transducers). If any machine computing $f$ satisfies (), then $f$ is aperiodic; conversely, if the minimal machine violates (*) for the state transformation of a string $v$, then two powers $\delta^i \neq \delta^j$ recur infinitely often, and minimality yields strings $u, w$ for which the last letters of $f(u v^i w)$ and $f(u v^j w)$ differ, contradicting aperiodicity. This lemma is what the book's decision procedure for aperiodicity rests on.