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Zapiski Nauchnykh Seminarov POMI, 2005, Volume 322, Pages 83–106
(Mi znsl395)
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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
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
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
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https://www.mathnet.ru/eng/znsl395 https://www.mathnet.ru/eng/znsl/v322/p83
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