chapter 9 for Theorem \[thm2.5\], we state necessary and sufficient conditions for the validity of the $\varphi$-expansion, which are different from those in Parry’s paper [@P2], Theorem \[thm2.1bis\] and Theorem \[thm2.1ter\].

We then use $\varphi$-expansions to study two interesting and related problems in sections \[section3\] and \[section4\]. When one applies the method of section \[section2\] to the dynamical system $\beta x+\alpha\mod1$, one obtains a symbolic shift which is entirely described by two strings $\ud{u}^\ab$ and $\ud{v}^\ab$ of symbols in a finite alphabet $\tA=\{0,\ldots,k-1\}$. The shift space is given by $$\label{1.1}
\BSigma(\ud{u}^\ab,\ud{v}^\ab)=\big\{\ud{x}\in\tA^{\Z_+}\colon
\ud{u}^\ab\preceq\sigma^n\ud{x}\preceq\ud{v}^\ab\;\,\forall n\geq
0 \big\}\,,$$ where $\preceq$ is the lexicographic order and $\sigma$ the shift map. The particular case $\alpha=0$ has been much studied from many different viewpoints ($\beta$-shifts). For $\alpha\not=0$ the structure of the shift space is richer. A natural problem is to study all shift spaces $\Sigma(\ud{u},\ud{v})$ of the form when we replace $\ud{u}^\ab$ and $\ud{v}^\ab$ by a pair of strings $\ud{u}$ and $\ud{v}$. In section \[section3\] we give an algorithm, Theorem \[thm3.1\], based on the $\varphi$-expansion, which allows to compute the topological entropy of shift spaces $\Sigma(\ud{u},\ud{v})$. One of the essential tool is the follower-set graph associated to the shift space. This graph is presented in details 