Endomorphisms of the shift dynamical system, discrete derivatives, and applications

All continuous endomorphisms f∞ of the shift dynamical system S on the 2-adic integers Z2 are induced by some f : Bn → {0, 1}, where n is a positive integer, Bn is the set of n-blocks over {0, 1}, and f∞ (x) = y0y1y2 . . . where for all i ∈ N, yi = f (xixi+1 . . . xi+n−1). Define D : Z2 → Z2 to be the endomorphism of S induced by the map {(00, 0) , (01, 1) , (10, 1) , (11, 0)} and V : Z2 → Z2 by V (x) = −1− x. We prove that D, V ◦D, S, and V ◦S are conjugate to S and are the only continuous endomorphisms of Swhose parity vector function is solenoidal. We investigate the properties of D as a dynamical system, and use D to construct a conjugacy from the 3x + 1 function T : Z2 → Z2 to a parity-neutral dynamical system. We also construct a conjugacy R from D to T . We apply these results to establish that, in order to prove the 3x + 1 conjecture, it suffices to show that for anym ∈ Z, there exists some n ∈ N such that R−1 (m) has binary representation of the form x0x1 . . . x2n−1 or x0x1x2 . . . x2n . © 2009 Elsevier B.V. All rights reserved.