Abstract
Combinatorics It is known that a Sturmian sequence S can be defined as a coding of the orbit of rho (called the intercept of S) under a rotation of irrational angle alpha (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.
Highlights
Sturmian sequences are infinite words over a binary alphabet with minimal complexity, and these sequences admit several equivalent definitions under different names
Which kind of Sturmian sequence can be fixed by certain non-trivial substitutions?
Yasutomi [10] gave a complete answer to the second question, by considering how the three elementary invertible substitutions change the slope and intercept of a Sturmian sequence
Summary
Sturmian sequences are infinite words over a binary alphabet with minimal complexity, and these sequences admit several equivalent definitions under different names. We know that, given a non-trivial invertible substitution τ , if ξ is a fixed point of τ (i.e. τ (ξ) = ξ), ξ is a Sturmian sequence (with the following exceptions of non-primitive substitutions: (01n, 1) which fixes 01∞, and (0, 10n) which fixes 10∞). Given an invertible substitution with a fixed point, what is the slope and the intercept?. Which kind of Sturmian sequence can be fixed by certain non-trivial substitutions?. Yasutomi [10] gave a complete answer to the second question, by considering how the three elementary invertible substitutions change the slope and intercept of a Sturmian sequence. We recall a characterization of the invertible substitution, and provide a unified treatment to the two questions. With help of the characterization, our proofs are very elementary and concise
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