Neighbours of self-affine tiles in lattice tilings

Let T be a tile of a self-affine lattice tiling. We give an algorithm that allows to determine all neighbours of T in the tiling. This can be used to characterize the sets V-L of points, where T meets L other tiles. Our algorithm generalizes an algorithm of the authors which was applicable only to a special class of self-affine lattice tilings. This new algorithm can also be applied to classes containing infinitely many tilings at once. Together with the results in recent papers by Bandt and Wang as well as Akiyama and Thuswaldner it allows to characterize classes of plane tilings which are homeomorphic to a disc. Furthermore, it sheds some light on the relations between different kinds of characterizations of the boundary of T.