Fiber denseness of intermediate β-shifts of finite type

Abstract

We focus on ( 1), and . The -expansions of critical point are called kneading invariants, denoted as . Let with being periodic, we state that 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 with its being a SFT is dense in . Similarly for the fiber . When considering another fiber , we demonstrate that when β is not a multinacci number, there are only countably many distinct matching intervals on . Using Markov approximation, we prove that the set of with being a SFT is dense in each matching interval. We also propose a classification scheme for the endpoints of these matching intervals.