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

All continuous endomorphisms f∞f∞ of the shift dynamical system SS on the 2-adic integers Z2Z2 are induced by some f:Bn→{0,1}, where nn is a positive integer, BnBn is the set of nn-blocks over {0, 1}, and f∞(x)=y0y1y2…f∞(x)=y0y1y2… where for all i∈Ni∈N, yi=f(xixi+1…xi+n−1)yi=f(xixi+1…xi+n−1). Define D:Z2→Z2D:Z2→Z2 to be the endomorphism of SS induced by the map {(00,0),(01,1),(10,1),(11,0)}{(00,0),(01,1),(10,1),(11,0)} and V:Z2→Z2V:Z2→Z2 by V(x)=−1−xV(x)=−1−x. We prove that DD, V∘DV∘D, SS, and V∘SV∘S are conjugate to SS and are the only continuous endomorphisms of SS whose parity vector function is solenoidal. We investigate the properties of DD as a dynamical system, and use DD to construct a conjugacy from the 3x+13x+1 function T:Z2→Z2T:Z2→Z2 to a parity-neutral dynamical system. We also construct a conjugacy RR from DD to TT. We apply these results to establish that, in order to prove the 3x+13x+1 conjecture, it suffices to show that for any m∈Z+m∈Z+, there exists some n∈Nn∈N such that R−1(m)R−1(m) has binary representation of the form x0x1…x2n−1¯ or x0x1x2…x2n¯.