Powers and alternative laws

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.