$\ud{u}^\ab$, resp. $\ud{v}^\ab$, $$\label{virtual}
\sigma^k\ud{u}^\ab=\lim_{x\downarrow
0}\i(T_\ab^k(x))\quad\text{and}\quad
\sigma^k\ud{v}^\ab=\lim_{x\uparrow 1}\i(T_\ab^k(x))\,.$$ Notice that $T^k_\ab(0)<1$ and $T^k_\ab(1)>0$ for all $k\geq 0$.

The virtual itineraries $\ud{u}\equiv\ud{u}^\ab$ and $\ud{v}\equiv\ud{v}^\ab$ of the dynamical system $\beta x+\alpha
\mod 1$ verify the conditions $$\label{condition}
\ud{u}\preceq\sigma^n\ud{u}\preceq \ud{v}\quad \forall\,n\geq
0\quad\text{and}\quad \ud{u}\preceq \sigma^n\ud{v}\preceq
\ud{v}\quad \forall\,n\geq 0\,.$$ By Theorem \[thm2.1ter\], and Theorem \[thm2.5\] we have $(x\in X\backslash S)$ $$\begin{aligned}
\ophi^\ab(\sigma^k\ud{u})&=\lim_{x\downarrow
0}\ophi^\ab(\i(T^k_\ab(x))=\lim_{x\downarrow 0}T^k_\ab(x)\equiv
T^k_\ab(0) \label{validity1}\\
\ophi^\ab(\sigma^k\ud{v})&=\lim_{x\uparrow
1}\ophi^\ab(\i(T^k_\ab(x))=\lim_{x\uparrow 1}T^k_\ab(x)\equiv
T^k_\ab(1)\,. \label{validity2}\end{aligned}$$ Hence $\ud{u}$ and $\ud{v}$ verify the equations[^6] $$\label{equation}
\ophi^\ab(\ud{u})=0\,,\;
\ophi^\ab(\sigma\ud{u})=\alpha\quad\text{and}\quad
\ophi^\ab(\ud{v})=1\,,\; \ophi^\ab(\sigma\ud{v})=\gamma\,,$$ with $$\label{gamma}
\gamma:=\alpha+\beta-k+1\in(0,1]\,.$$ The strings $\ud{u}^\ab$ and $\ud{v}^\ab$ are $\varphi$-expansions of $0$ and $1$. Because of the presence of discontinuities for the transformation $T_\ab$ at $a_1,\ldots
a_{k-1}$, there are other strings $\ud{u}$, $\ud{v}$ which verify and , and which are also $\varphi$-expansions of $0$ and $1$. For latter purposes we need to decribe these strings; this is the content of Proposition \[pro3.4\], Proposition \[pro3.4bis\] and Proposition \[pro3.4ter\]. We also take into consideration the borderline cases $\alpha=1$ and $\gamma=0$. When $\alpha=1$ or $\gamma=0$ the dynamical system $T_\ab$ is defined using formula . The orbits of $0$ and $1$ are defined as before. For example, if $\alpha=1$ it is the same dynamical system as $T_{0,\beta}$, but with different symbols for the coding of the 