- The 3x+1 (Collatz) conjecture
- Two mappings of Keith Matthews
- The 3x+371 (generalized-Collatz) conjecture
- A 3-branched generalized-Collatz conjecture

- Some of G. Venturini's mappings
- Two mappings of Mridul Sen
- John Horton Conway's cycling sequence
- Examples of H. Möller's mapping
- d-branched example of Walter Carnielli

- 3-branched example of Lu Pei

- d-branched example of Lu Pei

- b-branched example of Geoffrey Caveney

- d-branched example of Walter Carnielli
- Mappings of Benoit Cloitre

- Mapping of Kevin Lamoreau

- Mappings of Tomás Oliveira e Silva

- Frequency program
- Finding the canonical form of the Markov matrix Q
_{T}(m) arising from generalized 3x+1 mappings , where either gcd(m_{i}, d )=1 for 0 ≤ i < d, or gcd(m_{i}, d^{2}) = gcd(m_{i}, d) for 0 ≤ i < d and d divides m. - Finding the canonical form of Markov matrix Q
_{T}(m) arising from George Leigh's Markov chain {Y_{n}(x)}

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