 
Summary: On the connectedness of selfaĆne attractors
Shigeki AKIYAMA and Nertila GJINI y
Abstract
Let T = T (A; D) be a selfaĆne attractor in R n dened by an integral
expanding matrix A and a digit set D. In the rst part of this paper, in
connection with canonical number systems, we study connectedness of T when
D corresponds to the set of consecutive integers f0; 1; : : : ; j det(A)j 1g. It
is shown that in R 3 and 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 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 suĆcient condition of connectedness of the Pisot dual
tiles of degree 4. Detailed proofs will be given in [4].
1 Introduction
In this paper, we shall give a brief summary of the paper [4]. Proofs given here
are representative parts of detailed ones in [4]. Let Mn (Z) be the set of n n
