in subsection \[subsectionfollower\]. The algorithm is given in subsection \[subsectionalgo\] and the computations of the topological entropy in subsection \[topological\]. The basic idea of the algorithm is to compute two real numbers $\bar{\alpha}$ and $\bar{\beta}$, given the strings $\ud{u}$ and $\ud{v}$, and to show that the shift space $\BSigma(\ud{u},\ud{v})$ is a modification of the shift space $\Sigma(\ud{u}^\abb,\ud{v}^\abb)$ obtained from the dynamical system $\bar{\beta}x+\bar{\alpha}\mod1$, and that the topological entropies of the two shift spaces are the same. In the last section we consider the following inverse problem for the dynamical systems $\beta x+\alpha \mod1$: given $\ud{u}$ and $\ud{v}$, find $\alpha$ and $\beta$ so that $$\ud{u}=\ud{u}^\ab\quad\text{and}\quad\ud{v}=\ud{v}^\ab\,.$$ The solution of this problem is given in Theorems \[thm4.1\] and \[thm4.2\] for all $\beta>1$.

$\varphi$-expansion for piecewise monotone dynamical\
systems {#section2}
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Piecewise monotone dynamical systems {#subsection2.1}
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Let $X:=[0,1]$ (with the euclidean distance). We consider the case of piecewise monotone dynamical systems of the following type. Let $0=a_0<a_1<\cdots<a_k=1$ and $I_j:=(a_j,a_{j+1})$, $j\in\tA$. We set $\tA:=\{0,\ldots,k-1\}$, $k\geq 2$, and $$S_0:=X\backslash \bigcup_{j\in\tA}I_j\,.$$ For each $j\in\tA$ let $$f_j:I_j\mapsto J_j:=f_j(I_j)\subset [0,1]$$ be a strictly monotone continuous map. When necessary we also denote by $f_j$ the continuous extension of the map on the closure $\overline{I}_j$ of $I_j$. We define a 