On conjugacies of the 3x+13x+1 map induced by continuous endomorphisms of the shift dynamical system

Lagarias showed that the shift dynamical system SS on the set Z2Z2 of 2-adic integers is conjugate to the famous 3x+13x+1 map TT under a conjugacy ΦΦ. Thus for each continuous endomorphism f∞f∞ of SS there is a corresponding endomorphism Hf=Φ∘f∞∘Φ−1Hf=Φ∘f∞∘Φ−1 of TT and a map ΨfΨf from the coimage of HfHf to itself defined by Ψf([x])=[T(x)]Ψf([x])=[T(x)]. In this paper, we completely classify all continuous endomorphisms f∞f∞ of SS for which ΨfΨf is conjugate to TT. We then define an infinite family of such maps, ΨMkΨMk, that are “neutral” modulo 2k−12k−1 in the sense that each element of the domain is a complete residue system modulo 2k−12k−1. By investigating the relationships between TT-cycles and the ΨMkΨMk-cycles that contain them, we obtain an alternate method for studying the dynamics of TT. This method is used to prove several new results pertaining to TT-cycles, which are then applied to yield several possible approaches to the 3x+13x+1 conjecture.