Remarks on Self-Affine Tilings

Self-Affine Tiles in Rn

A self-affine tile in R is a set T of positive measure with A(T) = d ∈ $ < (T + d), where A is an expanding n × n real matrix with det (A) = m on integer, and $ = {d 1 ,d 2 , . . . , d m } ⊆ R is a set of m digits. It is known that self-affine tiles always give tilings of R by translation. This paper extends the known characterization of digit sets $ yielding self-affine tiles. It proves seve...

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 VL 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 c...

PSEUDO-SELF-AFFINE TILINGS IN Rd

It is proved that every pseudo-self-affine tiling in R is mutually locally derivable with a self-affine tiling. A characterization of pseudo-self-similar tilings in terms of derived Voronöı tessellations is a corollary. Previously, these results were obtained in the planar case, jointly with Priebe Frank. The new approach is based on the theory of graph-directed iterated function systems and su...

Some remarks on self-a ne tilings

We study self-aane tilings of R n with special emphasis on the two-digit case. We prove that in this case the tile is connected and, if n 3, is a lattice-tile.

On the Boundary of Self Affine Tilings Generated by Pisot Numbers