definition of the upper mechanical word ${\bf s}'(\gamma, \rho) = \left(s'_n(\gamma, \rho) \right)_{n \in \N}$ is analogous, it just uses the interval $I_0 = (0,1-\gamma]$. Let us stress that $s_n(\gamma, \rho) \neq s'_n(\gamma, \rho)$ for at most two neighbouring indices $n$ and $n+1$. All upper and lower mechanical words with irrational slope are Sturmian and any Sturmian word equals to a lower or to an upper mechanical word. Let us stress that one-sided Sturmian words with irrational slope are always uniformly recurrent. The language of a Sturmian word depends only on $\gamma$. The number $\gamma$ is in fact the density of the letter $1$, i.e., $\gamma = \lim\limits_{n\to \infty}\tfrac1n \# \left \{i\in \mathbb{N} \colon i < n, s_i(\gamma, \rho) = 1 \right \} $. Consequently, $1-\gamma$ is the density of the letter $0$.

For any irrational $\gamma \in (0,1)$ there exists a unique mechanical word ${\bf c}(\gamma)$ with slope $\gamma$ such that both $0{\bf c}(\gamma)$ and $1{\bf c}(\gamma)$ are Sturmian. The word ${\bf c}(\gamma)$ is a standard Sturmian word and ${\bf c}(\gamma) ={\bf s}(\gamma, \gamma) ={\bf s}'(\gamma, \gamma)$. Many further properties of Sturmian words can be found in [@Lo83; @BeSe_Lothaire].

For our study of derivated words, the following result of 