  Take an odd number x >0. Then 3x+1 is even and one can find an integer k> 0
so that y= 3x+1/2^k is again odd. We get in this way the mapping T, Tx=y. The
paper contains two theorems describing statistical properties of T. The first
describes statistical properties of trajectories for large x. The second gives
some information about the probability distribution appearing in this problem.
