 Positive-word growth under a free-group endomorphism

Let an endomorphism of the free group on $x,y$ fix $x$ and send $y$ to $yw$, where $w$ is a nonempty word containing only the positive letters $x,y$. Every iterate of $y$ is again a positive word containing $y$. Substitution increases its length by $|w|$ times the number of occurrences of $y$, hence by at least $|w|$. No cancellation can occur, so $|{\varphi }^n(y)|\ge 1+n|w|$.

The uniqueness of the reduced word in a free group now proves that no positive iterate of the endomorphism is the identity. This criterion does not require the endomorphism to be invertible; for example the replacement $y\mapsto y^2$ is included.