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.