Fiber denseness of intermediate β-shifts of finite type

摘要

We focus on T ss,a(x) = ss x+ a ( mod 1), x 2 [0, 1] and (ss, a) 2. := f (ss, a) 2 R2 : ss 2 (1, 2) and 0 < a < 2 ss g. The T +/- ss,a-expansions t +/- ss,a (x) of critical point c ss, a = 1- a ss are called kneading invariants, denoted as (k+, k-). Let.(k+) := f(ss, a) 2. : t + ss,a (c ss,a) = k+g with k+ being periodic, we state that.(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 (ss, a) with its O ss, a being a SFT is dense in.(k+). Similarly for the fiber.(k-). When considering another fiber.(ss) := f(ss, a) 2. : ss 2 (1, 2) is fixed g, we demonstrate that when ss is not a multinacci number, there are only countably many distinct matching intervals on.(ss). Using Markov approximation, we prove that the set of (ss, a) with O ss, a being a SFT is dense in each matching interval. We also propose a classification scheme for the endpoints of these matching intervals.

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