  In this paper, we consider the one-to-one correspondence between a 2-adic
integer and its parity sequence under iteration of the so-called "3x+1" map.
First, we prove a new formula for the inverse transform. Next, we briefly
review what is known about the induced automorphism and study its dynamics on
the 2-adic integers. We find that it is ergodic on many small odd invariant
sets, and that it has two odd cycles of period 2 in addition to its two odd
fixed points. Finally, a plane embedding is presented, for which we establish
affine self-similarity by using functional equations.
