Intermediate β-shifts of finite type

Abstract

An aim of this article is to highlight dynamical differences between the greedy, and hence the lazy, $β$-shift (transformation) and an intermediate $β$-shift (transformation), for a fixed $β\in (1, 2)$. Specifically, a classification in terms of the kneading invariants of the linear maps $T_{β,α} \colon x \mapsto βx + α\bmod 1$ for which the corresponding intermediate $β$-shift is of finite type is given. This characterisation is then employed to construct a class of pairs $(β,α)$ such that the intermediate $β$-shift associated with $T_{β, α}$ is a subshift of finite type. It is also proved that these maps $T_{β,α}$ are not transitive. This is in contrast to the situation for the corresponding greedy and lazy $β$-shifts and $β$-transformations, for which both of the two properties do not hold.