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Zapiski Nauchnykh Seminarov POMI, 2005, Volume 322, Pages 83–106 (Mi znsl395)  

This article is cited in 57 scientific papers (total in 57 papers)

Rauzy tilings and bounded remainder sets on the torus

V. G. Zhuravlev

Vladimir State Pedagogical University
References:
Abstract: For the two dimensional torus $\mathbb{T}^2$ we construct the Rauzy tilings $d^0\supset d^1\supset\ldots\supset d^m\supset\ldots$, where each tiling $d^{m+1}$ turns out by inflation of $d^{m}$. The following results are proved:
1) Any tiling $d^{m}$ is invariant with respect to the shift $S(x)=x+\begin{pmatrix} \zeta \\ \zeta ^2\end{pmatrix}\mod\mathbb{Z}^2$, here $\zeta^{-1}> 1$ is a Pisot number satisfying the equation $x^3-x^2-x-1=0$.
2) The induced map $S^{(m)}=S|_{B^m d}$ is an exchange transformation of $B^m d\subset\mathbb{T}^2$, where $d=d^0$ and $B=\begin{pmatrix} - \zeta & - \zeta \\ 1-\zeta ^2 & \zeta^2\end{pmatrix}$.
3) The map $S^{(m)}$ is a shift of the torus $B^m d\simeq\mathbb{T}^2$ and $S^{(m)}$ is isomorphic to the initial shift $S$. It means that $d^m$ are infinite differentiable tilings.
Let $Z_N(X)$ be equal to the number of points in the orbit $S^1(0), S^2(0)$, $\ldots,S^N(0)$ visited the domain $B^m d$. Then the remainder $r_N(B^md)=Z_N(B^m d)-N\zeta^m$ satisfies $-1.7<r_N(B^m d)<0.5$ for all $m$.
Received: 05.03.2005
English version:
Journal of Mathematical Sciences (New York), 2006, Volume 137, Issue 2, Pages 4658–4672
DOI: https://doi.org/10.1007/s10958-006-0262-z
Bibliographic databases:
UDC: 519.68
Language: Russian
Citation: V. G. Zhuravlev, “Rauzy tilings and bounded remainder sets on the torus”, Proceedings on number theory, Zap. Nauchn. Sem. POMI, 322, POMI, St. Petersburg, 2005, 83–106; J. Math. Sci. (N. Y.), 137:2 (2006), 4658–4672
Citation in format AMSBIB
\Bibitem{Zhu05}
\by V.~G.~Zhuravlev
\paper Rauzy tilings and bounded remainder sets on the torus
\inbook Proceedings on number theory
\serial Zap. Nauchn. Sem. POMI
\yr 2005
\vol 322
\pages 83--106
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl395}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2138453}
\zmath{https://zbmath.org/?q=an:1158.11331}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2006
\vol 137
\issue 2
\pages 4658--4672
\crossref{https://doi.org/10.1007/s10958-006-0262-z}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33746121532}
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  • https://www.mathnet.ru/eng/znsl/v322/p83
  • This publication is cited in the following 57 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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