|
This article is cited in 42 scientific papers (total in 42 papers)
One-dimensional Fibonacci tilings
V. G. Zhuravlev Vladimir State Pedagogical University
Abstract:
We use the $B$-operator to construct a family of Fibonacci tilings
$\operatorname{Til}(\varepsilon_m)$ of the unit interval $I_0=[0,1)$
consisting of $F_{m+1}$ short and $F_{m+2}$ long elementary intervals
with the ratio of the lengths equal to the golden section
$\tau=\frac{1+\sqrt{5}}2$. We prove that the tilings
$\operatorname{Til}(\varepsilon_m)$ satisfy a recurrence relation similar
to the relation $F_{m+2}=F_{m+1}+F_m$ for the Fibonacci numbers.
The ends
of the elementary intervals in the tilings $\operatorname{Til}(\varepsilon_m)$
form a sequence of points $O_0$ whose derivatives
$d^mO_0 = O_0 \cap [1-\tau^{-m},1)$
are sequences $O_m$ similar to the
sequence $O_0$. We compute the direct $R_m(i)$ and inverse $R_{-m}(i)$
renormalizations for the sequences $O_m$. We establish a connection between
our tilings and the Sturm sequence, and give some applications of the tilings
$\operatorname{Til}(\varepsilon_m)$ in the theory of numbers.
Received: 19.11.2002 Revised: 28.02.2004
Citation:
V. G. Zhuravlev, “One-dimensional Fibonacci tilings”, Izv. RAN. Ser. Mat., 71:2 (2007), 89–122; Izv. Math., 71:2 (2007), 307–340
Linking options:
https://www.mathnet.ru/eng/im620https://doi.org/10.1070/IM2007v071n02ABEH002358 https://www.mathnet.ru/eng/im/v71/i2/p89
|
Statistics & downloads: |
Abstract page: | 995 | Russian version PDF: | 339 | English version PDF: | 17 | References: | 75 | First page: | 4 |
|