On the connectedness of self-affine attractors

Abstract.

Let T = T(A, D) be a self-affine attractor in $\mathbb{R}^n$ defined by an integral expanding matrix A and a digit set D. In the first part of this paper, in connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers $\{0, 1,\ldots, |\det(A)| - 1\}$ . It is shown that in $\mathbb{R}^3$ and $\mathbb{R}^4$ , for any integral expanding matrix A, T(A, D) is connected.

In the second part, we study connectedness of Pisot dual tiles, which play an important role in the study of $\beta$ -expansions, substitutions and symbolic dynamical systems. It is shown that each tile of the dual tiling generated by a Pisot unit of degree 3 is arcwise connected. This is naturally expected since the digit set consists of consecutive integers as above. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4. We even give a simple necessary and sufficient condition of connectedness of the Pisot dual tiles of degree 4. Detailed proofs will be given in [4].