Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (6): 1921-1937.doi: 10.1007/s10473-021-0609-z

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

Yuke HUANG1, Zhiying WEN2   

  1. 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).

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

CLC Number: 

  • 68R15
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