Intermediate β-shifts of finite type

Abstract

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

v3: 19 pages, 6 figures, fixed typos and minor errors, to appear in Discrete Contin. Dyn. Syst. A