Coding of irrational rotation Sturmian word

enlarge animation showing sturmian sequence generated irrational rotation θ≈0.2882 , x≈0.0789

for



θ
∈
[
0
,
1
)


{\displaystyle \theta \in [0,1)}

, define




t

θ


:
[
0
,
1
)
→
[
0
,
1
)


{\displaystyle t_{\theta }:[0,1)\to [0,1)}





t
↦
t
+
θ

mod


1


{\displaystyle t\mapsto t+\theta \mod 1}

.



x
∈
[
0
,
1
)


{\displaystyle x\in [0,1)}

define θ-coding of x sequence (xn) where

x

n


=

{




1



 if 


t

θ


n


(
x
)
∈
[
0
,
θ
)




0



 else









{\displaystyle x_{n}=\left\{{\begin{array}{cl}1&{\text{ if }}t_{\theta }^{n}(x)\in [0,\theta )\\0&{\text{ else}}\end{array}}\right.}

.

let w infinite sequence of 0s , 1s. sequence w sturmian if



x
∈
[
0
,
1
)


{\displaystyle x\in [0,1)}

, irrational



θ
∈
(
0
,
∞
)


{\displaystyle \theta \in (0,\infty )}

, w θ-coding of x.