The 3x + 1 conjugacy map

Abstract

The 3x + 1 map T and the shift map S are defined by T(x) = (3x + 1)/2 for x odd, T(x) = x/2 for x even, while S(x) = (x - 1)/2 for x odd, S(x) = x/2 for x even. The 3x + 1 conjugacy map Φ on the 2-adic integers Z2 conjugates S to T, i.e., Φ ○ S ○ Φ-1 = T. The map Φ mod 2n induces a permutation Φn on Z/2nZ. We study the cycle structure of Φn. In particular we show that it has order 2n-4 for n ≥ 6.We also count 1-cycles of Φn for n up to 1000; the results suggest that Φ has exactly two odd fixed points. The results generalize to the ax + b map, where ab is odd.