DERIVED SEQUENCES AND THE FACTOR SPECTRUM OF THE PERIOD-DOUBLING SEQUENCE

doi:10.1007/s10473-021-0609-z

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (6): 1921-1937.doi: 10.1007/s10473-021-0609-z

DERIVED SEQUENCES AND THE FACTOR SPECTRUM OF THE PERIOD-DOUBLING SEQUENCE

黄煜可¹, 文志英²

1. School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China;
2. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

收稿日期:2021-03-29 修回日期:2021-07-21 出版日期:2021-12-25 发布日期:2021-12-27
通讯作者: Yuke HUANG,E-mail:hyk@bupt.edu.cn E-mail:hyk@bupt.edu.cn
作者简介:Zhiying WEN,E-mail:wenzy@tsinghua.edu.cn
基金资助:
The first author was supported by National Natural Science Foundation of China (11701024), the Fundamental Research Funds for the Central Universities (2019RC17).

DERIVED SEQUENCES AND THE FACTOR SPECTRUM OF THE PERIOD-DOUBLING SEQUENCE

Yuke HUANG¹, Zhiying WEN²

1. School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China;
2. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received:2021-03-29 Revised:2021-07-21 Online:2021-12-25 Published:2021-12-27
Supported by:
The first author was supported by National Natural Science Foundation of China (11701024), the Fundamental Research Funds for the Central Universities (2019RC17).

摘要/Abstract

摘要： Factor properties and their structures are important themes in combinatorics on words. Let $\mathbb{D}$ be the infinite one-sided sequence over the alphabet $\{a,b\}$ generated by the period-doubling substitution $\sigma(a)=ab$ and $\sigma(b)=aa$. In this paper, we determine the derived sequence $D_w$($\mathbb{D}$) for any factor ω $\prec$ $\mathbb{D}$, and study some factor spectra using the structures of derived sequences. We also prove the reflexivity property of derived sequences.

关键词: combinatorics on words, envelope word, derived sequence, factor spectrum, reflexivity of derived sequence

Abstract: Factor properties and their structures are important themes in combinatorics on words. Let $\mathbb{D}$ be the infinite one-sided sequence over the alphabet $\{a,b\}$ generated by the period-doubling substitution $\sigma(a)=ab$ and $\sigma(b)=aa$. In this paper, we determine the derived sequence $D_w$($\mathbb{D}$) for any factor ω $\prec$ $\mathbb{D}$, and study some factor spectra using the structures of derived sequences. We also prove the reflexivity property of derived sequences.

Key words: combinatorics on words, envelope word, derived sequence, factor spectrum, reflexivity of derived sequence

中图分类号:

68R15

黄煜可, 文志英. DERIVED SEQUENCES AND THE FACTOR SPECTRUM OF THE PERIOD-DOUBLING SEQUENCE[J]. 数学物理学报(英文版), 2021, 41(6): 1921-1937.

Yuke HUANG, Zhiying WEN. DERIVED SEQUENCES AND THE FACTOR SPECTRUM OF THE PERIOD-DOUBLING SEQUENCE[J]. Acta mathematica scientia,Series B, 2021, 41(6): 1921-1937.

参考文献 23

[1]	Allouche J P, Baake M, Cassaigne J, Damanik D. Palindrome complexity. Theor Comput Sci, 2003, 292(1):9-31
[2]	Allouche J P, Peyri$mathbb{D}$re J, Wen Z X, Wen Z Y. Hankel determinants of the thue-morse sequence. AnnalesInstitut Fourier, 1998, 1(1):1-27
[3]	Allouche J P, Shallit J. Automatic Sequences:Theory, Applications, Generalizations. Cambridge:Cambridge University Press, 2003
[4]	Charlier É, Rampersad N, Shallit J. Enumeration and decidable proper ties of automatic sequences. Int J Found Comput S, 2012, 23(05):1035-1066
[5]	Durand F. A characterization of substitutive sequences using return words. Discrete Math, 1998, 179:89-101
[6]	Damanik D. Singular continuous spectrum for the period doubling hamiltonian on a set of full measure. Commun Math Phys, 1998, 196(2):477-483
[7]	Damanik D. Local symmetries in the period-doubling sequence. Discrete Appl Math, 2000, 100(1/2):115- 121
[8]	Damanik D. Uniform singular continuous spectrum for the period doubling hamiltonian. Ann Henri Poincaré, 2001, 2(1):101-108
[9]	Fu H, Han G N. On t-extensions of the Hankel determinants of certain automatic sequences. Theor Comput Sci, 2015, 562(C):46-56
[10]	Fokkink R J, Kraaikamp C, Shallit J. Hankel matrices for the period-doubling sequence. Indagat Math New Ser, 2017, 28(1):108-119
[11]	Guo Y J, Wen Z X. Automaticity of the Hankel determinants of difference sequences of the Thue-morse sequence. Theor Comput Sci, 2014, 552(4):1-12
[12]	Huang Y K, Han G N, Wen Z Y. Derived sequences and enumeration problems of the (n,j)-bonacci sequence. 2019.06. Preprint
[13]	Huang Y K, Wen Z Y. The sequence of return words of the Fibonacci sequence. Theor Comput Sci, 2015, 593:106-116
[14]	Huang Y K, Wen Z Y. Kernel words and gap sequence of the Tribonacci sequence. Acta Math Sci, 2016, 36B(1):173-194
[15]	Huang Y K, Wen Z Y. The numbers of repeated palindromes in the Fibonacci and Tribonacci sequences. Discrete Appl Math, 2017, 230:78-90
[16]	Huang Y K, Wen Z Y. The factor spectrum and derived sequence. Journal of Mathematical Research with Applications, 2019, 39(6):718-732
[17]	Huang Y K, Zhang H X, Gap sequence of cutting sequence with slope θ=[0;d]. arXiv:1408.3724, 2014
[18]	Liviotti E. A study of the structure factor of Thue-morse and period-doubling chains by wavelet analysis. J Phys-Condens Mat, 1998, 8(27):5007
[19]	Lotheaire M. Combinatorics on words//Encyclopedia of Mathematics and Its Applications 17. Addisonweslry, 1997
[20]	Lothaire M. Algebraic Combinatorics on words//Encyclopedia of Mathematics and Its Applications 90. Cambridge University Press, 2002
[21]	Lothaire M. Applied Combinatorics on words//Encyclopedia of Mathematics and Its Applications 105. Cambridge University Press, 2005
[22]	Makarov M. On the infinite permutation generated by the period doubling word. Eur J Combin, 2010, 31(1):368-378
[23]	Wen Z X, Wen Z Y. Some properties of the singular words of the Fibonacci word. Eur J Combin, 1994, 15:587-598

DERIVED SEQUENCES AND THE FACTOR SPECTRUM OF THE PERIOD-DOUBLING SEQUENCE

DERIVED SEQUENCES AND THE FACTOR SPECTRUM OF THE PERIOD-DOUBLING SEQUENCE

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献 23

相关文章 0

Metrics

本文评价

推荐阅读 10