On Christoffel and standard words and their derivatives

Abstract

We introduce and study natural derivatives for Christoffel and standard words, as well as for characteristic Sturmian words. These derivatives, which are defined as inverse images under suitable morphisms, preserve the aforementioned classes of words. In the case of Christoffel words, the morphisms involved map $a$ to $a^{k+1}b$ (resp., $ab^k$) and $b$ to $a^kb$ (resp.,$ab^{k+1}$) for a suitable $k>0$. As long as derivatives are not just a single letter, higher-order derivatives are naturally obtained. We define the depth of a Christoffel or of a standard word as the smallest order for which the derivative is a single letter. We give several combinatorial and arithmetic descriptions of the depth, and (tight) lower and upper bounds for it

Extremal values of semi‐regular continuants and codings of interval exchange transformations

“…Since Christoffel words are exactly the images of $$a,b$$ under compositions of $$\lambda _a$$ and $$\rho _b$$ (see, e.g., [6, 12]), an alternative proof of Theorem 3 can easily be obtained from Lemmas 3.12, 3.13.…”

Extremal values of semi-regular continuants and codings of interval exchange transformations

“…Since Christoffel words are exactly the images of a, b under compositions of λ a and ρ b (see for instance [6,12]), an alternative proof of Theorem 4 can easily be obtained from Lemmas 3.12 and 3.13.…”