Geodesic
The 3x + 1 Conjugacy Map
Canadian journal of mathematics, Tome 48 (1996) no. 6, pp. 1154-1169

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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 Z 2 conjugates S to T, i.e., Φ o S o Φ-1 = T. The map Φ mod 2n induces a permutation Φn on Z/2n Z. We study the cycle structure of Φn . In particular we show that it has order 2 n − 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.
DOI : 10.4153/CJM-1996-060-x
Mots-clés : 11B75 
Bernstein, Daniel J.; Lagarias, Jeffrey C. The 3x + 1 Conjugacy Map. Canadian journal of mathematics, Tome 48 (1996) no. 6, pp. 1154-1169. doi: 10.4153/CJM-1996-060-x
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