Rauzy tilings and bounded remainder sets on the torus

For the two-dimensional torus double-struck T sign2, we construct the Rauzy tilings d0 ⊃ d1 ⊃|... ⊃ dm ⊃ ..., where each tiling dm+1 is obtained by subdividing the tiles of dm. The following results are proved. Any tiling dm is invariant with respect to the torus shift S(x) = x+ (ζζ2) mod ℤ2, where ζ-1 > 1 is the Pisot number satisfying the equation x3- x 2-x-1 = 0. The induced map S(m) = S|_Bmd is an exchange transformation of Bmd ⊂ double-struck T sign2, where d = d0 and B = (1 - ζ2 ζ 2 - ζ - ζ) . The map S(m) is a shift of the torus Bmd ≃ double-struck T sign2, which is affinely isomorphic to the original shift S. This means that the tilings dm are infinitely differentiable. If ZN(X) denotes the number of points in the orbit S1(0), S2(0), SN(0) belonging to the domain Bmd, then, for all m, the remainder rN(Bmd) = ZN(Bmd) - N ζm satisfies the bounds -1.7 < rN(Bmd) < 0.5. Bibliography: 10 titles. © 2006 Springer Science+Business Media, Inc.