Rauzy tilings and bounded remainder sets on the torus

Abstract

For the two-dimensional torus $$\mathbb{T}^2 $$ , 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+ $$\left( {_{\zeta ^2 }^\zeta } \right)$$ mod ℤ2, where ζ−1 > 1 is the Pisot number satisfying the equation x3− x2−x-1 = 0. The induced map $$S^{(m)} = \left. S \right|_{B^m d} $$ is an exchange transformation of Bmd ⊂ $$\mathbb{T}^2 $$ , where d = d0 and $$ B = \left( {_{1 - \zeta ^2 \zeta ^2 }^{ - \zeta - \zeta } } \right) $$ . The map S(m) is a shift of the torus $$B^m d \simeq \mathbb{T}^2 $$ , 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.