are treated similarly.

Dynamical system $\beta x+\alpha\mod 1$ {#subsection2.3}
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We consider the family of dynamical systems $\beta x+\alpha\mod 1$ with $\beta >1$ and $0\leq \alpha<1$. For given $\alpha$ and $\beta$, the dynamical system is described by $k=\lceil
\alpha+\beta\rceil$ intervals $I_j$ and maps $f_j$, $$I_0=\Big(0,\frac{1-\alpha}{\beta}\Big)\,,\,I_j=\Big(\frac{j-\alpha}{\beta},
\frac{j+1-\alpha}{\beta}\Big)
\,,\,j=1,\ldots,k-2\,,\,I_{k-1}=\Big(\frac{k-1-\alpha}{\beta},1\Big)$$ and $$f_j(x)=\beta x+\alpha-j\,,\,j=0,\ldots k-1\,.$$ The maps $T_{\alpha,\beta}$, $\varphi^\ab$ and $\overline{\varphi}^\ab$ are defined as in subsection \[subsection2.1\]. $$\overline{\varphi}^\ab(x)=\begin{cases} 0 & \text{if
$0\leq x\leq\alpha$}\\
\beta^{-1}(x-\alpha) & \text{if $\alpha\leq x\leq \alpha+\beta$} \\
1 & \text{if $\alpha+\beta\leq x\leq \lceil \alpha+\beta\rceil$}
\end{cases}$$ and $$\label{eqso}
S_0=\{a_j\colon j=1,\ldots,k-1\}\cup\{0,1\}\quad\text{with}\quad
a_j:=\beta^{-1}(j-\alpha)\,.$$ Since all maps are increasing the total order on $\tA^{\Z_+}$ is the lexicographic order. We have $2k$ virtual orbits, but only two of them are important. Indeed, if we set $$\ud{u}^{\alpha,\beta}:=\ud{u}^0\quad\text{and}
\quad\ud{v}^{\alpha,\beta}:=\ud{v}^{k-1}\,,$$ then $$\ud{u}^j=j\ud{u}^{\alpha,\beta}\,,\,j=1,\ldots k-1$$ and $$\ud{v}^j=j\ud{v}^{\alpha,\beta}\,,\,j=0,\ldots,k-2\,.$$

\[pro3.1\] Let $\beta >1$ and $0\leq \alpha<1$. The $\varphi$-expansion for the dynamical system $\beta x+\alpha\mod1$ is valid. $$\Sigma^{\alpha,\beta}:=\big\{\i(x)\in\tA^{\Z_+}\colon x\in
X\backslash S\big\}=\big\{\ud{x}\in\tA^{\Z_+}\colon
\ud{u}^{\alpha,\beta}\prec\sigma^n\ud{x}\prec\ud{v}^{\alpha,\beta}\quad\forall
n\geq 0\big\}\,.$$ Moreover $$\ud{u}^{\alpha,\beta}\preceq\sigma^n\ud{u}^{\alpha,\beta}\prec\ud{v}^{\alpha,\beta}
\quad\text{and}\quad
\ud{u}^{\alpha,\beta}\prec\sigma^n\ud{v}^{\alpha,\beta}\preceq\ud{v}^{\alpha,\beta}
\quad\forall n\geq0\,.$$

The closure of $\Sigma^{\alpha,\beta}$ is the shift space $$\label{3.2}
\BSigma(\ud{u}^{\alpha,\beta},\ud{v}^{\alpha,\beta}):=\big\{\ud{x}\in\tA^{\Z_+}\colon
\ud{u}^{\alpha,\beta}\preceq\sigma^n\ud{x}\preceq\ud{v}^{\alpha,\beta}\quad\forall
n\geq 0\big\}\,.$$ We define the [orbits of $0$, resp. $1$]{} as, (the limits are taken with $x\in X\backslash S$) $$T^k_\ab(0):=\lim_{x\downarrow 0}T_\ab^k(x)\,,\,k\geq
0\quad\text{resp.}\quad T^k_\ab(1):=\lim_{x\uparrow
1}T_\ab^k(x)\,,\,k\geq 0\,.$$ From and the coding of these orbits is 