On the Powers of the Collatz Function

Abstract

For all natural numbers a, b and $d > 0$, we consider the function $f_{a,b,d}$ which associates n/d with any integer n when it is a multiple of d, and $an + b$ otherwise; in particular $f_{3,1,2}$ is the Collatz function. To realize these functions by transducers (automata labelled by pairs of words), the coding in reverse base 2 is generally used. For the Collatz function, it gives a simple 5-states transducer but it is not suitable for the composition and so far, no one has been able to specify, for all integers p, a generic transducer computing its composition p  times. Coding in direct base ad with $b < a$, we realize the functions $f_{a,b,d}$ by synchronous sequential transducers. This particular form makes explicit the composition of such a transducer p times to compute $f^p_{a,b,d}$ in terms of p and a, b, d. We even give an explicit construction of an infinite transducer realizing the closure under composition of $f_{a,b,d}$.