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 Z 2 are induced by some f : B n → { 0 , 1 } , where n is a positive integer, B n is the set of n -blocks over {0, 1}, and f ∞ ( x ) = y 0 y 1 y 2 … where for all i ∈ N , y i = f ( x i x i + 1 … x i + n − 1 ) . Define D : Z 2 → Z 2 to be the endomorphism of S induced by the map { ( 00 , 0 ) , ( 01 , 1 ) , ( 10 , 1 ) , ( 11 , 0 ) } and V : Z 2 → Z 2 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 S whose parity vector function is solenoidal. We investigate the properties of D as a dynamical system, and use D to construct a conjugacy from the 3 x + 1 function T : Z 2 → Z 2 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 3 x + 1 conjecture, it suffices to show that for any m ∈ Z + , there exists some n ∈ N such that R − 1 ( m ) has binary representation of the form x 0 x 1 … x 2 n − 1 ¯ or x 0 x 1 x 2 … x 2 n ¯ .