The Collatz conjecture and De Bruijn graphs

We study variants of the well-known Collatz graph, by considering the action of the 3n+13n+1 function on congruence classes. For moduli equal to powers of 2, these graphs are shown to be isomorphic to binary De Bruijn graphs. Unlike the Collatz graph, these graphs are very structured, and have several interesting properties. We then look at a natural generalization of these finite graphs to the 2-adic integers, and show that the isomorphism between the resulting infinite graphs is exactly the conjugacy map previously studied by Bernstein and Lagarias. Finally, we show that for generalizations of the 3n+13n+1 function, such as the family of an+ban+b functions and Collatz-like functions, we get similar relations with 2-adic and pp-adic De Bruijn graphs respectively.