Real $3x+1$

The famous $3x+1$ problem involves applying two maps: $T_0(x)=x/2$ and $T_1(x)=(3x+1)/2$ to positive integers. If $x$ is even, one applies $T_0$, if it is odd, one applies $T_1$. The conjecture states that each trajectory of the system arrives to the periodic orbit $\{1,2\}$. In this paper, instead of choosing each time which map to apply, we allow ourselves more freedom and apply both $T_0$ and $T_1$independently of $x$. That is, we consider the action of the free semigroup with generators $T_0$ and $T_1$ on the space of positive real numbers. We prove that this action is minimal (each trajectory is dense) and that the periodic points are dense. Moreover, we give a full characterization of the group of transformations of the real line generated by $T_0$ and $T_1$.