On conjugacies of the map induced by continuous endomorphisms of the shift dynamical system

Lagarias showed that the shift dynamical system S on the set Z 2 of 2-adic integers is conjugate to the famous 3 x + 1 map T under a conjugacy Φ . Thus for each continuous endomorphism f ∞ of S there is a corresponding endomorphism H f = Φ ∘ f ∞ ∘ Φ − 1 of T and a map Ψ f from the coimage of H f to itself defined by Ψ f ( [ x ] ) = [ T ( x ) ] . In this paper, we completely classify all continuous endomorphisms f ∞ of S for which Ψ f is conjugate to T . We then define an infinite family of such maps, Ψ M k , that are “neutral” modulo 2 k − 1 in the sense that each element of the domain is a complete residue system modulo 2 k − 1 . By investigating the relationships between T -cycles and the Ψ M k -cycles that contain them, we obtain an alternate method for studying the dynamics of T . This method is used to prove several new results pertaining to T -cycles, which are then applied to yield several possible approaches to the 3 x + 1 conjecture.