A Collatz-type conjecture on the set of rational numbers

Let θ(x)=(x−1)/3 if x⩾1, and θ(x)=2x/(1−x) if 0⩽x<1. We conjecture that the θ-orbit of every nonnegative rational number ends in 0. A weaker conjecture asserts that there are no positive rational fixed points for any map in the semigroup Λ generated by the maps 3x+1 and x/(x+2). In this paper, we prove that the asymptotic density of the set of maps in Λ that have rational fixed points is zero. Moreover, we prove that certain types of elements in the semigroup Λ cannot have rational fixed points.