On substitution Invariant Sturmian Words: An Application of Rauzy Fractals

Sturmian words are infinite words that have exactly n + 1 factors of length n for every positive integer n. A Sturmian word s�,� is also defined as a coding over a two-letter alphabet of the orbit of the pointunder the action of the irrational rotation R� : x 7! x + � (mod 1). Yasutomi characterized in (34) all the pairs (�,�) such that the Sturmian word s�,� is a fixed point of some non-trivial substitution. By investigating the Rauzy frac- tals associated with invertible substitutions, we give an alternative geometric proof of Yasutomi's characterization.