Discrete Mathematics & Theoretical Computer Science, Vol 13, No 2 (2011)

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Sturmian Sequences and Invertible Substitutions

Li Peng, Bo Tan


It is known that a Sturmian sequence S can be defined as a coding of the orbit of ρ (called the intercept of S) under a rotation of irrational angle α (called the slope). On the other hand, a fixed point of an invertible substitution is Sturmian. Naturally, there are two interrelated questions: (1) Given an invertible substitution, we know that its fixed point is Sturmian. What is the slope and intercept? (2) Which kind of Sturmian sequences can be fixed by certain non-trivial invertible substitutions? In this paper we give a unified treatment to the two questions. We remark that though the results are known, our proof is very elementary and concise.

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