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Induced bounded remainder sets

Author: V. G. Zhuravlev
Translated by: A. Luzgarev
Original publication: Algebra i Analiz, tom 28 (2016), nomer 5.
Journal: St. Petersburg Math. J. 28 (2017), 671-688
MSC (2010): Primary 52C20; Secondary 51M20
Published electronically: July 25, 2017
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Abstract: The induced two-dimensional Rauzy tilings are generalized to tiling of the tori $ \mathbb{T}^D= \mathbb{R}^D/ \mathbb{Z}^D$ of arbitrary dimension $ D$. For that, a technique of embedding $ T\stackrel {\operatorname {em}}{\hookrightarrow } \mathbb{T}^D$ of toric developments $ T$ into the torus $ \mathbb{T}^D_L = \mathbb{R}^D/ L$ for some lattice $ L$ is used. A feature of the developments $ T$ is that for a given shift $ S: \mathbb{T}^D \longrightarrow \mathbb{T}^D$ of the torus, its restriction $ S\vert _T$ to the subset $ T \subset \mathbb{T}^D$, i.e., the first recurrence map, or the Poincaré map, is equivalent to an exchange transformation of the tiles $ T_k$ that form a tiling of the development $ T=T_0\sqcup T_1\sqcup \dots \sqcup T_D$. In the case under consideration, the induced map $ S\vert _T$ is a translation of the torus $ \mathbb{T}^D_L$.

It is proved that every $ T_k$ is a bounded remainder set: the deviations $ \delta _{T_k}(i,x_{0})$ in the formula $ r_{T_k}(i,x_{0})= a_{T_k} i + \delta _{T_k}(i,x_{0})$ are bounded, where $ r_{T}(i,x_{0})$ is the number of occurrences of the points $ S^{0}(x_{0}), S^{1}(x_{0}),\dots , S^{i-1}(x_{0})$ from the $ S$-orbit in the set $ T_k$, $ x_0$ is an arbitrary starting point on the torus $ \mathbb{T}^D$, and the coefficient $ a_{T_k}$ equals the volume of $ T_k$. Explicit estimates are obtained for these deviations $ \delta _{T_k}(i,x_{0})$. Earlier, the relationship between the maps $ S\vert _T$ and bounded remainder sets was noticed by Rauzy and Ferenczi.

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V. G. Zhuravlev
Affiliation: Vladimir State University, pr. Stroiteley 11, 600024 Vladimir, Russia

Keywords: Poincar\'e map, bounded remainder sets
Received by editor(s): November 1, 2014
Published electronically: July 25, 2017
Additional Notes: Financially supported by RSF, project no. 14-01-00360
Article copyright: © Copyright 2017 American Mathematical Society