## Denjoy systems and substitutions.(English)Zbl 1247.37013

Let $$\mathcal{A}=\{0,\dots,d\}$$ be an alphabet and $$\mathcal{A}^\star$$ the free monoid over $$\mathcal{A}$$ with the empty word $$\varepsilon_\emptyset$$ as identity element. The reversal of a finite word $$w= w_1\cdots w_n$$ is given by $$\overleftarrow{w}= w_n \cdots w_1$$. A substitution $$\sigma$$ over $$\mathcal{A}$$ is a map $$\mathcal{A} \to \mathcal{A}^\star \setminus \{\varepsilon_\emptyset\}$$ and can be extended to a morphism of $$\mathcal{A}^\star$$ naturally. The reversal substitution $$\overleftarrow{\sigma}$$ is given by $$\overleftarrow{\sigma}(i) := \overleftarrow{\sigma(i)}$$ for all $$i \in \mathcal{A}$$.
Let $$(\sigma_1,\sigma_2,\dots)$$ be any infinite sequence of substitutions. This sequence generates a right infinite word $$w=w_1w_2\cdots$$ if $w=\lim_{n \to \infty} \sigma_1\sigma_2 \cdots \sigma_n(i) \qquad \text{for all $$i \in \mathcal{A}$$}.$ The sequence $$(\sigma_n)_n$$ generates the bi-infinite word $$w=\cdots w_{-1}w_0w_1\cdots$$ if $$(\sigma_n)_n$$ generates $$w_0w_1\cdots$$ and $$(\overleftarrow{\sigma_n})_n$$ generates $$w_{-1}w_{-2}\cdots$$.
The paper considers the following coding of an irrational rotation. Let $$S^1=\mathbb{R}/\mathbb{Z}$$ and $$R_\alpha:S^1 \to S^1$$ be the rotation $$R_\alpha(z) = z+\alpha$$ modulo $$\mathbb{Z}$$ (with $$\alpha \in \mathbb{R}$$). We identify $$S^1$$ with $$[0,1)$$ and consider a finite ordered sequence $$\Lambda= \{\omega_1< \dots <\omega_{d_1}\}$$ with $$\omega_i \in (0,1)$$. $$\Lambda$$ induces the partition $$t_0(0) =[0,\omega_0)$$, $$t_0(i)=[\omega_{i-1},\omega_i)$$ for $$i=1,\dots,d$$ with $$\omega_d=1$$. We have the following map: $J:S^1 \to \mathcal{A}^\mathbb{Z}, \omega \mapsto \big(J(\omega)_n\big)_{n \in \mathbb{Z}}$ with $$J(\omega)_n = i \iff R_\alpha^n(\omega) \in t_0(i)$$.
The paper generalizes the following “folklore” proposition for two-digit alphabets to $$d$$-digit alphabets using techniques based on Denjoy systems, Bratteli-Vershik systems and substitution systems.
Proposition: Take $$d=1$$ and $$\mathcal{A}=\{0,1\}$$ the two-digit alphabet. Let $$\alpha = [0,a_1,a_2,\dots]$$ be the (regular) infinite continued fraction of the irrational $$\alpha \in [0,1)$$. Put $$\sigma_n(0)=0\underbrace{1 \cdots 1}_{a_n \text{ times}}$$ and $$\sigma_n(1)=0\underbrace{1 \cdots 1}_{a_n-1 \text{ times}}$$ for each $$n \in \mathbb{Z}$$. Then the sequence $$(\sigma_1,\overleftarrow{\sigma}_2\sigma_3,\overleftarrow{\sigma}_4, \dots)$$ generates $$J_\alpha(\alpha)$$.

### MSC:

 37B10 Symbolic dynamics 05A05 Permutations, words, matrices
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### References:

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