PERIODICITY OF THE UNIVOQUE β-EXPANSIONS

doi:10.1016/S0252-9602(16)30113-8

数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (1): 33-46.doi: 10.1016/S0252-9602(16)30113-8

PERIODICITY OF THE UNIVOQUE β-EXPANSIONS

戈跃华^1,2, 谭波¹

1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China;
2. School of Mathematics and Physics, Hebei GEO University, Shijiazhuang 050031, China

收稿日期:2016-01-07 修回日期:2016-05-05 出版日期:2017-02-25 发布日期:2017-02-25
通讯作者: Bo TAN,E-mail:tanbo@mail.hust.edu.cn E-mail:tanbo@mail.hust.edu.cn
作者简介:Yuehua GE,E-mail:geyuehua1001@126.com;
基金资助:
This work was supported by NSFC (11171123, 11222111).

PERIODICITY OF THE UNIVOQUE β-EXPANSIONS

Yuehua GE^1,2, Bo TAN¹

1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China;
2. School of Mathematics and Physics, Hebei GEO University, Shijiazhuang 050031, China

Received:2016-01-07 Revised:2016-05-05 Online:2017-02-25 Published:2017-02-25
Contact: Bo TAN,E-mail:tanbo@mail.hust.edu.cn E-mail:tanbo@mail.hust.edu.cn
About author:Yuehua GE,E-mail:geyuehua1001@126.com;
Supported by:
This work was supported by NSFC (11171123, 11222111).

摘要/Abstract

摘要：

Let m≥1 be an integer, 1 < β < m+1. A sequence ε1ε2ε3… with ε_i∈{0, 1, …,m} is called a β-expansion of a real number x if x=∑_i(ε_i)/(βⁱ). It is known that when the base β is smaller than the generalized golden ration, any number has uncountably many expansions, while when β is larger, there are numbers which has unique expansion. In this paper, we consider the bases such that there is some number whose unique expansion is purely periodic with the given smallest period. We prove that such bases form an open interval, moreover, any two such open intervals have inclusion relationship according to the Sharkovskiǐ ordering between the given minimal periods. We remark that our result answers an open question posed by Baker, and the proof for the case m=1 is due to Allouche, Clarke and Sidorov.

关键词: beta-expansions, periodic expansions, unique expansion, the Sharkovskii ordering

Abstract:

Key words: beta-expansions, periodic expansions, unique expansion, the Sharkovskii ordering

戈跃华, 谭波. PERIODICITY OF THE UNIVOQUE β-EXPANSIONS[J]. 数学物理学报(英文版), 2017, 37(1): 33-46.

Yuehua GE, Bo TAN. PERIODICITY OF THE UNIVOQUE β-EXPANSIONS[J]. Acta mathematica scientia,Series B, 2017, 37(1): 33-46.

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PERIODICITY OF THE UNIVOQUE β-EXPANSIONS

PERIODICITY OF THE UNIVOQUE β-EXPANSIONS

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