 
Summary: ON THE TOPOLOGICAL STRUCTURE OF FRACTAL TILINGS
GENERATED BY QUADRATIC NUMBER SYSTEMS
SHIGEKI AKIYAMA AND JšORG M. THUSWALDNER
Abstract. Let be a root of an irreducible quadratic polynomial x2 +Ax+B
with integer coefficients A, B and assume that forms a canonical number
system, i.e. each x Z[] admits a representation of the shape
x = a0 + a1 + · · · + ahh
with ai {0, 1, . . . , B  1}. It is possible to attach a tiling to such a number
system in a natural way. If 2A < B +3 then we show that the fractal boundary
of the tiles of this tiling is a simple closed curve and its interior is connected.
Furthermore, the exact set equation for the boundary of a tile is given. If
2A B + 3 then the topological structure of the tiles is quite involved. In this
case we prove that the interior of a tile is disconnected. Furthermore, we are
able to construct finite labelled directed graphs which allow to determine the set
of "neighbours" of a given tile T , i. e. the set of all tiles which have nonempty
intersection with T . In a next step we give the structure of the set of points,
in which T coincides with L other tiles. In this paper we use two different
approaches: Geometry of numbers and finite automata theory. Each of these
approaches has its advantages and allows to emphasize on different properties
of the tiling. Especially the conjecture in AkiyamaThuswaldner [1] that for
