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A SURVEY ON TOPOLOGICAL PROPERTIES OF TILES RELATED TO NUMBER SYSTEMS
 

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 self-affine
tiles. After reviewing some basic results on self-affine 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 self-affine 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 non-empty 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 non-empty compact set T which
satisfies

  

Source: Akiyama, Shigeki - Department of Mathematics, Niigata University

 

Collections: Mathematics