Odometers on Regular Languages

Odometers or "adding machines" are usually introduced in the context of positional numeration systems built on a strictly increasing sequence of integers. We generalize this notion to systems defined on an arbitrary infinite regular language. In this latter situation, if (A,<) is a totally ordered alphabet, then enumerating the words of a regular language L over A with respect to the induced genealogical ordering gives a one-to-one correspondence between ℕ and L. In this general setting the odometer is not defined on a set of sequences of digits but on a set of pairs of sequences where the first (resp. the second) component of the pair is an infinite word over A (resp. an infinite sequence of states of the minimal automaton of L). We study some properties of the odometer such as continuity, injectivity, surjectivity, minimality, .... We then study some particular cases: we show the equivalence of this new function with the classical odometer built upon a sequence of integers whenever the set of greedy representations of all the integers is a regular language; we also consider substitution numeration systems as well as the connection with β-numerations.