Subshifts of finite type and matching for intermediate $\beta$-transformations

We focus on the relationships between matching and subshift of finite type for intermediate $\beta$-transformations $T_{\beta ,\alpha }(x)=\beta x+\alpha$ ($$ 1), where $x\in [0,1]$ and $(\beta ,\alpha )\in \mathrm{\Delta }:=\{(\beta ,\alpha )\in {\mathbb{R}}^2:\beta \in (1,2)\mathrm{a}\mathrm{n}\mathrm{d}0<\alpha <2-\beta \}$. We prove that if the kneading space ${\mathrm{\Omega }}_{\beta ,\alpha }$ is a subshift of finite type, then $T_{\beta ,\alpha }$ has matching. Moreover, each $(\beta ,\alpha )\in \mathrm{\Delta }$ with $T_{\beta ,\alpha }$ has matching corresponds to a matching interval, and there are at most countable different matching intervals on the fiber. Using combinatorial approach, we construct a pair of linearizable periodic kneading invariants and show that, for any $\epsilon >0$ and $(\beta ,\alpha )\in \mathrm{\Delta }$ with $T_{\beta ,\alpha }$ has matching, there exists $(\beta ,{\alpha }^{\mathrm{\prime }})$ on the fiber with $\mathrm{\mid }\alpha -{\alpha }^{\mathrm{\prime }}\mathrm{\mid }<\epsilon$, such that ${\mathrm{\Omega }}_{\beta ,{\alpha }^{\mathrm{\prime }}}$ is a subshift of finite type. As a result, the set of $(\beta ,\alpha )$ for which ${\mathrm{\Omega }}_{\beta ,\alpha }$ is a subshift of finite type is dense on the fiber if and only if the set of $(\beta ,\alpha )$ for which $T_{\beta ,\alpha }$ has matching is dense on the fiber.