Asymmetric tent map expansions. II: Purely periodic points
type: Item
instance of: scholarly article
title: Asymmetric tent map expansions. II: Purely periodic points
zbMATH Open document ID: 0809.11042
published in: Illinois Journal of Mathematics
publication date: 1994-08-23T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11K55
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 37E99
zbMATH DE Number: 622559
zbMATH Keywords: points with terminating expansion
zbMATH Keywords: asymmetric tent map
zbMATH Keywords: purely periodic points
zbMATH Keywords: Pisot number
zbMATH Keywords: hyperbolic iterated function system
author: Jeffrey C. Lagarias
author: Horacio Porta
author: Kenneth B. Stolarsky
reviewed by: Robert F. Tichy
MaRDI profile type: Publication
Recommended article: Asymmetric Tent Map Expansions. I. Eventually Periodic Points
Recommended article: Dynamical properties of the tent map
Recommended article: Periodicity ofβ-expansions for certain Pisot units
Recommended article: On periodic \(\beta\)-expansions of Pisot numbers and Rauzy fractals
Recommended article: Q3509581
Recommended article: Q4385018
Recommended article: Dynamics forβ-shifts and Diophantine approximation
Recommended article: On purely periodic beta-expansions of Pisot numbers
Recommended article: Beta-expansion of 1 for quartic Pisot units
Recommended article: The beta-transformation's companion map for Pisot or Salem numbers and their periodic orbits
description: scientific article; zbMATH DE number 622559