@article{BernsteinLagarias1996,
  author    = {{Daniel J. Bernstein and Jeffrey C. Lagarias}},
  title     = {{The $3x+1$ Conjugacy Map}},
  journal   = {{Canadian J. Math.}},
  year      = {{1996}},
  volume    = {{48}},
  pages     = {{1154--1169}},
  doi       = {{10.4153/CJM-1996-060-x}},
  url       = {{https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/3x-1-conjugacy-map/6975BB4A8C46CF6842217043AAF9EC13}},
  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 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/2^n 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.}},
  note      = {{MR 98a:11027}},
}