Thin homogeneous sets of factors

Abstract

This paper provides an algorithm to decide whether a set of words of length n is exactly the set of factors of length n of a unique bi-infinite word. In case of positive answer, this set of factors is said thin. We prove that a bi-infinite word u admits a thin set of factors of some length n iff u is periodic or ultimately periodic on the left and on the right but not with the same period. In other respects, as a tool for the proof, we give a standard form to the writing of a rational bi-infinite word which allows us to count easily its number of factors of length n, (i.e. : a words u such that {u} is the set recognized by a finite automation).