The authors consider the intermediate \(\beta\)-shifts originating from the intermediate \(\beta\)-transformations \(T_{\beta,\alpha}^\pm\) defined from the two-parameter family of maps \(T_{\beta,\alpha}:[0,1] \rightarrow [0,1]\) where \(T_{\beta,\alpha}(x)= \beta x + \alpha {\textrm{ (mod }} 1)\) with the parameter space \[\Delta= \{ (\beta,\alpha) \in \mathbb{R}^2: \beta \in (1,2) {\textrm{ and }} 0<\alpha<2-\beta \}.\] The union of the images of \([0,1]\) under the \(T_{\beta,\alpha}^\pm\)-expansions \(\tau_{\beta,\alpha}^\pm(x)\) of \(x \in [0,1]\) is the kneading space \(\Omega_{\beta,\alpha}\) and the kneading invariants of \(\Omega_{\beta,\alpha}\) is the pair of sequences \((k_+, k_- ) = (\tau_{\beta,\alpha}^+(c_{\beta,\alpha}), \tau_{\beta,\alpha}^-(c_{\beta,\alpha}))\) for the critical point \(c_{\beta,\alpha} = (1 - \alpha)/\beta\); \(\Omega_{\beta,\alpha}\) is a subshift.
B. Li et al. [Proc. Am. Math. Soc. 147, No. 5, 2045–2055 (2019; Zbl 1442.37027)] proved that the set of \(\Delta\)-parameters such that \(\Omega_{\beta,\alpha}\) is a subshift of finite type is dense in \(\Delta\). In the paper here, the authors fix a periodic \(k_+\) and define \(\Delta(k_+)\) to be the set of \(\Delta\)-parameters such that \(\tau_{\beta,\alpha}^+(c_{\beta,\alpha})=k_+\). One of their main results is that the set of \(\Delta\)-parameters such that \(\Omega_{\beta,\alpha}\) is a subshift of finite type is dense in the fiber \(\Delta(k_+)\). Similarly, the set of \(\Delta\)-parameters such that \(\Omega_{\beta,\alpha}\) is a subshift of finite type is dense in the fiber \(\Delta(k_-)\). These findings are similar to a result of W. Parry [Acta Math. Acad. Sci. Hung. 11, 401–416 (1960; Zbl 0099.28103)] showing that the set of \(\beta\)’s such that \(\Omega_{\beta,0}\) is a subshift of finite type is dense in \((1,2)\).
B. Quackenbush et al. [Mathematics 8, No. 6, 903–919 (2020; doi:10.3390/math8060903)] proved for a fixed multinacci number \(\beta \in (1,2)\) that the set of parameters such that \(\Omega_{\beta,\alpha}\) is a subshift of finite type is dense in \(\Delta(\beta)=\{(\beta,\alpha)\}\). The authors of the present paper show that if \(\beta\) is not a multinacci number then there are atmost countably many subintervals of \(\Delta(\beta)\) such that the set of \((\beta,\alpha)\)’s where \(\Omega_{\beta,\alpha}\) is a subshift of finite type is dense in each subinterval.