Complexity of infinite words associated with beta-expansion

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F04%3A04105324" target="_blank" >RIV/68407700:21340/04:04105324 - isvavai.cz</a>

Result on the web

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DOI - Digital Object Identifier

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Result language

čeština

Original language name

Complexity of infinite words associated with beta-expansion

Original language description

We study the complexity of the infinite word $u_beta$ associated with the R'enyi expansion of $1$ in an irrational base $beta>1$. When $beta$ is the golden ratio, this is the well known Fibonacci word, which is Sturmian, and of complexity $C(n)=n+1$. For $beta$ such that $d_beta(1)=t_1t_2cdots t_{m}$ is finite we provide a simple description of the structure of special factors of the word $u_beta$. When $t_m=1$ we show that $C(n)=(m-1)n+1$. In the cases when $t_1=t_2=cdots=t_{m-1}$ or $t_1>max{t_2,dots,t_{m-1}}$ we show that the first difference of the complexity function $C(n+1)-C(n)$ takes value in ${m-1,m}$ for every $n$, and consequently we determine the complexity of $u_beta$. We show that $u_beta$ is an Arnoux-Rauzy sequenceif and only if $d_beta(1)=t,tcdots,t,1$. On the example of $beta=1+2cos(2pi/7)$, solution of $X^3=2X^2+X-1$, we illustrate that the structure of special factors is more complicated for $d_beta(1)$ infinite even

Czech name

Complexity of infinite words associated with beta-expansion

Czech description

We study the complexity of the infinite word $u_beta$ associated with the R'enyi expansion of $1$ in an irrational base $beta>1$. When $beta$ is the golden ratio, this is the well known Fibonacci word, which is Sturmian, and of complexity $C(n)=n+1$. For $beta$ such that $d_beta(1)=t_1t_2cdots t_{m}$ is finite we provide a simple description of the structure of special factors of the word $u_beta$. When $t_m=1$ we show that $C(n)=(m-1)n+1$. In the cases when $t_1=t_2=cdots=t_{m-1}$ or $t_1>max{t_2,dots,t_{m-1}}$ we show that the first difference of the complexity function $C(n+1)-C(n)$ takes value in ${m-1,m}$ for every $n$, and consequently we determine the complexity of $u_beta$. We show that $u_beta$ is an Arnoux-Rauzy sequenceif and only if $d_beta(1)=t,tcdots,t,1$. On the example of $beta=1+2cos(2pi/7)$, solution of $X^3=2X^2+X-1$, we illustrate that the structure of special factors is more complicated for $d_beta(1)$ infinite even

Type

J_x - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)

CEP classification

BE - Theoretical physics

OECD FORD branch

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Project

<a href="/en/project/GA201%2F01%2F0130" target="_blank" >GA201/01/0130: Some aspects of quantum group and self-similar aperiodic structures</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2004

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

RAIRO-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE

ISSN

0764-583X

e-ISSN

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Volume of the periodical

38

Issue of the periodical within the volume

2

Country of publishing house

FR - FRANCE

Number of pages

23

Pages from-to

162-184

UT code for WoS article

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EID of the result in the Scopus database

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