 
Summary: A SURVEY ON TOPOLOGICAL PROPERTIES OF TILES RELATED TO
NUMBER SYSTEMS
SHIGEKI AKIYAMA AND JšORG M. THUSWALDNER
Abstract. In the present paper we give an overview of topological properties of selfaffine
tiles. After reviewing some basic results on selfaffine tiles and their boundary we give cri
teria for their local connectivity and connectivity. Furthermore, we study the connectivity
of the interior of a family of tiles associated to quadratic number systems and give results
on their fundamental group. If a selfaffine tile tessellates the space the structure of the set
of its "neighbors" is discussed.
1. Introduction and basic definitions
Let X be a complete metric space and let fi : X X (1 i m) be injective
contractions. In [31] it is proved that there is a unique compact nonempty set K satisfying
K = f1(K) . . . fm(K).
{fi}1im is called iterated function system (IFS for short). K is called the attractor of this
IFS.
Let A be an expanding d Ś d matrix (i.e. a matrix each of whose eigenvalues is strictly
greater than 1) and suppose that  det(A) = m for some integer m > 1. Let D :=
{a1, . . . , am} Rd
be a finite set of vectors. Then the nonempty compact set T which
satisfies
