Asymmetric tent map expansions. II: Purely periodic points (Q1331502)

This paper is a continuation of the first part [J. Lond. Math. Soc. 47, 542-556 (1993)] reviewed above. In the present paper the authors consider again the \(T_ \alpha\)-expansion of numbers \(x\in [0,1]\), where \(T_ \alpha\) is an asymmetric tent map \((\alpha>1)\). They study the sets \(\text{Fix}(T_ \alpha)\) of all purely periodic points and the set \(\text{Per}_ 0 (T_ \alpha)\) of all points with terminating \(T_ \alpha\)-expansion, when \(\alpha\) is a special Pisot number, i.e. \(\beta= {\alpha\over {\alpha-1}}\) is a Pisot number, too. One of the main results is: (1) \(\text{Fix} (T_ \alpha)\subseteqq \{\gamma\in \mathbb{Q}(\alpha)\): \(\gamma\in [0,1]\) and \(\sigma(\gamma)\in A_ \alpha^ \sigma\) for all embeddings \(\sigma: \mathbb{Q}(\alpha)\to\mathbb{C}\) with \(\sigma(\alpha)\neq \alpha\}\), where \(A_ \alpha^ \sigma\) is a compact subset of \(\mathbb{C}\) which is the attractor of a certain hyperbolic iterated function system. Furthermore it is examined when equality holds. For \(\alpha=2\) it is shown that \(\text{Fix}(T_ 2)= \{{p\over q}\): \(2\nmid q\) and \(2\mid p\) and \(0\leq p<q\}\). However, whenever \(\alpha\) is a special Pisot number generating either a real quadratic or a non-totally real cubic field, equality holds in (1). In the final section related results for \(\text{Per}_ 0 (T_ \alpha)\) are established.