The 3 X + 1 Conjugacy Map

The 3x + 1 conjugacy map phi : Z sub 2 -> Z sub 2 on the 2- adic integers Z sub 2 conjugates the 2-adic shift map S to the 3x + 1 map T, where T(x) = sup (3x + 1) over 2 if x = 1 (mod 2,) T(x) = x over 2 if x = O (mod 2), i.e. Phi o S o Phi sup (-1) = T. The 3x + 1 conjecture is known to be equivalent to the conjecture that N improper subset Phi (sup 1 over 3 Z). The map Phi (mod 2 sup n) induces a permutation bar over Phi sub n on Z/2 sup n Z and this paper studies the cycle structure of bar over Phi sub n. In particular it is shown that it has order 2 sup (n-4) for n => 6. The set of one-cycles of bar over Phi sub n is studied numerically for n = 950, and some (heuristic) inferences are drawn about possible nonzero fixed points of Phi.