  The subshift of finite type property (also known as the Markov property) is
ubiquitous in dynamical systems and the simplest and most widely studied class
of dynamical systems are $\beta$-shifts, namely transformations of the form
$T_{\beta, \alpha} \colon x \mapsto \beta x + \alpha \bmod{1}$ acting on
$[-\alpha/(\beta - 1), (1-\alpha)/(\beta - 1)]$, where $(\beta, \alpha) \in
\Delta$ is fixed and where $\Delta = \{ (\beta, \alpha) \in \mathbb{R}^{2}
\colon \beta \in (1,2) \; \text{and} \; 0 \leq \alpha \leq 2-\beta \}$.
Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045-2055,
2019), that the set of $(\beta, \alpha)$ such that $T_{\beta, \alpha}$ has the
subshift of finite type property is dense in the parameter space $\Delta$.
Here, they proposed the following question. Given a fixed $\beta \in (1, 2)$
which is the $n$-th root of a Perron number, does there exists a dense set of
$\alpha$ in the fiber $\{\beta\} \times (0, 2- \beta)$, so that $T_{\beta,
\alpha}$ has the subshift of finite type property?
  We answer this question in the positive for a class of Pisot numbers.
Further, we investigate if this question holds true when replacing the subshift
of finite type property by the property of beginning sofic (that is a factor of
a subshift of finite). In doing so we generalise, a classical result of Schmidt
(Bull. London Math. Soc., 12(4): 269-278, 1980) from the case when $\alpha = 0$
to the case when $\alpha \in (0, 2 - \beta)$. That is, we examine the structure
of the set of eventually periodic points of $T_{\beta, \alpha}$ when $\beta$ is
a Pisot number and when $\beta$ is the $n$-th root of a Pisot number.
