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

Abstract

All continuous endomorphisms f[subscript ∞] of the shift dynamical system S on the 2-adic integers Z[subscript 2] are induced by some f : B[subscript n]→{0,1}, where n is a positive integer, B[subscript n] is the set of n-blocks over {0, 1}, and f[subscript ∞](x)=y[subscript 0]y[subscript 1]y[subscript 2]…f[subscript ∞](x) = y[subscript 0]y[subscript 1]y[subscript 2]… where for all i∈N, yi = f(x[subscript i]x[subscript i+1]…x[subscript i+n−1]). Define D:Z[subscript 2]→Z[subscript 2] to be the endomorphism of S induced by the map {(00,0),(01,1),(10,1),(11,0)} and V:Z[subscript 2]→Z[subscript 2] by V(x)=−1−x. We prove that D, V∘DV∘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 3x+1 function T:Z[subscript 2]→Z[subscript 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 3x+1 conjecture, it suffices to show that for any m∈Z[superscript +], there exists some n∈N such that R[superscript −1](m) has binary representation of the form [bar over x[subscript 0]x[subscript 1]…x[subscript 2n−1]] or [bar over x[subscript 0]x[subscript 1]x[subscript 2]…x[subscript 2n]].