  A groupoid is alternative if it satisfies the alternative laws $x(xy)=(xx)y$
and $x(yy)=(xy)y$. These laws induce four partial maps on $\mathbb{N}^+\times
\mathbb{N}^+$, $(r,\,s)\mapsto (2r,\,s-r)$, $(r-s,\,2s)$, $(r/2,\,s+r/2)$,
$(r+s/2,\,s/2)$ that taken together form a dynamical system. We describe the
orbits of this dynamical system, which allows us to show that $n$th powers in a
free alternative groupoid on one generator are well-defined if and only if
$n\le 5$. We then discuss some number theoretical properties of the orbits, and
the existence of alternative loops without two-sided inverses.
