Sturmian Sequences and Invertible Substitutions †

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.


Introduction
Sturmian sequences are infinite words over a binary alphabet with minimal complexity, and these sequences admit several equivalent definitions under different names.In this paper, we adopt the following definition.
Sturmian sequences are extensively studied by many authors, and excellent descriptions can be found in Chapter 2 of [5] by J. Berstel & P. Séébold, and in Chapter 6 of [6] by P. Arnoux.|ε| = 0), and |u| i denotes the number of occurrences of the letter i in the word u.The mirror of a word A morphism τ : A * → A * is called a substitution of A * .Two substitutions τ and σ are said to be conjugate, if there is a word w ∈ A * such that τ (i)w = wσ(i) for any i ∈ A; or vice versa.
If a substitution τ can be extended to an automorphism of F (A), we say that it is invertible.In particular, an invertible substitution τ is non-erasing, that is, both τ (0) and τ (1) are different from ε.
A substitution τ can also be extended to a mapping of A N .And τ is called Sturmian if τ (ξ) is a Sturmian sequence for any Sturmian sequence ξ.Mignosi & Séébold [7] proved that a Sturmian substitution is also a composition of the above three substitutions.Therefore, a substitution is Sturmian if and only if it is invertible.
A substitution is called non-trivial if it is not the identity.From the above results, we know that, given a non-trivial invertible substitution τ , if ξ is a fixed point of τ (i.e.τ (ξ) = ξ), then ξ is a Sturmian sequence (with the following exceptions of non-primitive substitutions: (01 n , 1) which fixes 01 ∞ , and (0, 10 n ) which fixes 10 ∞ ).
Naturally, we ask the following two interrelated questions: QUESTION 1.Given an invertible substitution with a fixed point, what is the slope and the intercept?QUESTION 2. Which kind of Sturmian sequence can be fixed by certain non-trivial substitutions?
The first question was tackled in Tan & Wen [8].In fact, the slope is just the ratio of the frequencies of the two letters in the fixed point, and it is easy to determine via the substitution matrix.And the intercept is obtained by a very delicate comparison of the intercepts between the substitutions in a same conjugate class.
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.Later, Baláži, Masáková & Pelantová [1] and Berthé, Ei, Ito & Rao [2] gave alternative proofs of the characterization via the cutand-project scheme and the Rauzy fractal respectively.These proofs are somewhat technical and lengthy.
In this paper, we recall a characterization of the invertible substitution, and then provide a unified treatment to the two questions.With help of the characterization, our proofs are very elementary and concise.

Auxiliary Results
The shift function is the mapping T : And the shift changes the intercept of a Sturmian sequence in an obvious way.Lemma 2.1 We have that T S α,ρ = S α,α+ρ , T S α,ρ = S α,α+ρ .
We mention several special cases: S α,α = S α,α , called the characteristic sequence and also denoted by The next lemma is also readily checked.
The following characterization of invertible substitutions is shown in [8], which is essentially equivalent to the geometrical representation [4].The reader is referred to these papers for more details.3 The slope and intercept of a substitutive Sturmian sequence Let τ be an invertible substitution with a fixed point ξ.The substitution matrix of τ is defined as M = (|τ (j)| i ) i,j=0,1 .We will assume that M is primitive, and its Perron-Frobenius eigenvalue is denoted by λ.The slope and intercept of ξ are denoted by α and ρ respectively.In this section, given an invertible substitution τ , we calculate the slope α and intercept ρ.It is easy to determine α via the substitution matrix M .
The following theorem on the intercept is due to [8] (under a big guise).
Theorem 3.2 Let τ be a primitive invertible substitution with a fixed point.With the above notations, we have that where P (u) = (|u| 0 , |u| 1 ) T is the Parikh vector of u; λ is the Galois conjugate of λ; and u is the word associated with τ as in Theorem 2.3.

In general, if
for some K ∈ Z + , the same argument (with τ K in place of τ ) shows that η is a fixed point of some substitution in the conjugate class of τ K .So it suffices to show Putting K = m − m, we have and this concludes the proof of the sufficiency.2