Synchronizable functions on integers

Abstract: For all natural numbers a,b and d > 0, we consider the function f_{a,b,d} which associates n/d to any integer n when it is a multiple of d, and an + b otherwise; in particular f_{3,1,2} is the Collatz function. Coding in base a > 1 with b < a, we realize these functions by input-deterministic letter-to-letter transducers with additional output final words. This particular form allows to explicit, for any integer n, the composition n times of such a transducer to compute f^n_{a,b,d}. We even realize the closure under composition f^*_{a,b,d by an infinite input-deterministic letter-to-letter transducer with a regular set of initial states and a length recurrent terminal function.