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 ane self-similarity by using functional equations.
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