### Sturmian Sequences and Invertible Substitutions

*Li Peng, Bo Tan*

#### Abstract

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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