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.

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