Fiber denseness of intermediate β-shifts of finite type

Abstract

We focus on T β , α ( x ) = β x + α ( m o d 1), x ∈ [ 0 , 1 ] and ( β , α ) ∈ Δ := { ( β , α ) ∈ R 2 : β ∈ ( 1 , 2 ) and 0 < α < 2 − β } . The T β , α ± -expansions τ β , α ± ( x ) of critical point c β , α = 1 − α β are called kneading invariants, denoted as ( k + , k − ) . Let Δ ( k + ) := { ( β , α ) ∈ Δ : τ β , α + ( c β , α ) = k + } with k + being periodic, we state that Δ ( k + ) is a smooth curve which can be regarded as a fiber. By combinatorial method, we extend the results of Parry (1960 Acta Math. Acad. Sci. Hung. 11 401–16) and show that, the set of ( β , α ) with its Ω β , α being a SFT is dense in Δ ( k + ) . Similarly for the fiber Δ ( k − ) . When considering another fiber Δ ( β ) := { ( β , α ) ∈ Δ : β ∈ ( 1 , 2 ) is fixed } , we demonstrate that when β is not a multinacci number, there are only countably many distinct matching intervals on Δ ( β ) . Using Markov approximation, we prove that the set of ( β , α ) with Ω β , α being a SFT is dense in each matching interval. We also propose a classification scheme for the endpoints of these matching intervals.