Subshifts of finite type and matching for intermediate $β$-transformations
We focus on the relationships between matching and subshift of finite type for intermediate $β$-transformations $T_{β,α}(x)=βx+α$ ($\bmod$ 1), where $x\in[0,1]$ and $(β,α) \in Δ:= \{ (β, α) \in \mathbb{R}^{2}:β\in (1, 2) \; \rm{and} \; 0 < α<2 - β\}$. We prove that if the kneading space $Ω_{β,α}$ is a subshift of finite type, then $T_{β,α}$ has matching. Moreover, each $(β,α)\inΔ$ with $T_{β,α}$ 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 $ε>0$ and $(β,α)\inΔ$ with $T_{β,α}$ has matching, there exists $(β,α^{\prime})$ on the fiber with $|α-α^{\prime}|<ε$, such that $Ω_{β,α^{\prime}}$ is a subshift of finite type. As a result, the set of $(β,α)$ for which $Ω_{β,α}$ is a subshift of finite type is dense on the fiber if and only if the set of $(β,α)$ for which $T_{β,α}$ has matching is dense on the fiber.