  Define $\theta(x)=(x-1)/3$ if $x\geq 1$, and $\theta(x)=2x/(1-x)$ if $x<1$.
We conjecture that the orbit of every positive rational number ends in 0. In
particular, there does not exist any positive rational fixed point for a map in
the semigroup $\Omega$ generated by the maps $3x+1$ and $x/(x+2)$. In this
paper, we prove that the asymptotic density of the set of elements in $\Omega$
that have rational fixed points is zero.
