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 ∈ (1, 2). Specifically, a classification in terms of the kneading invariants of the linear maps Tβ,α : x ↦ βx + α mod 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 Tβ,α is a subshift of finite type. It is also proved that these maps Tβ,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.
|Number of pages||21|
|Journal||Discrete and Continuous Dynamical Systems|
|Publication status||Published - 1 Jun 2016|
- subshifts of finite type
- kneading theory