On conjugacies of the 3x+1 map induced by continuous endomorphisms of the shift dynamical system

Lagarias showed that the shift dynamical system S on the set Z2 of 2-adic integers is conjugate to the famous 3x + 1 map T under a conjugacy Φ . Thus for each continuous endomorphism f∞ of S there is a corresponding endomorphism Hf = Φ ◦ f∞ ◦ Φ−1 of T and a map Ψf from the coimage of Hf to itself defined by Ψf ([x]) = [T (x)]. In this paper, we completely classify all continuous endomorphisms f∞ of S for which Ψf is conjugate to T . We then define an infinite family of such maps, ΨMk , that are ‘‘neutral’’ modulo 2 k−1 in the sense that each element of the domain is a complete residue system modulo 2k−1. By investigating the relationships between T -cycles and theΨMk -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 3x+ 1 conjecture. © 2010 Elsevier B.V. All rights reserved.