Induced bounded remainder sets
Владимир Георгиевич Журавлев · St Petersburg Mathematical Journal · 2017
The induced two-dimensional Rauzy tilings are generalized to tiling of the tori T D = R D /Z D of arbitrary dimension D. For that, a technique of embedding T em ↩→ T D of toric developments T into the torus T D L = R D /L for some lattice L is used.A feature of the developments T is that for a given shift S : T D -→ T D of the torus, its restriction S| T to the subset T ⊂ 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 T 1 • • • T D .In the case under consideration, the induced map S| T is a translation of the torus) from the S-orbit in the set T k , x 0 is an arbitrary starting point on the torus T D , and the coefficient a T k equals the volume of T k .Explicit estimates are obtained for these deviations δ T k (i, x 0 ).Earlier, the relationship between the maps S| T and bounded remainder sets was noticed by Rauzy and Ferenczi.