Error diffusion on simplices: The structure of bounded invariant tiles

This is a companion paper to Adleret al. (in press, 2015). There, we proved the existence of an absorbing invariant tile for the Error Diffusion dynamics on an acute simplex when the input is constant and “ergodic” and we discuss the geometry of this tile. Under the same assumptions we prove here that said invariant tile (a fundamental set of the lattice generated by the vertices of the simplex) which is a finite union of polytopes have the property that any union of the intersections of the tile with the Voronoï regions of the vertices is a tile for a different, explicitly defined lattice.