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ON THE TOPOLOGICAL STRUCTURE OF FRACTAL TILINGS GENERATED BY QUADRATIC NUMBER SYSTEMS
 

Summary: ON THE TOPOLOGICAL STRUCTURE OF FRACTAL TILINGS
GENERATED BY QUADRATIC NUMBER SYSTEMS
SHIGEKI AKIYAMA AND J 
ORG M. THUSWALDNER
Abstract. Let be a root of an irreducible quadratic polynomial x 2 +Ax+B
with integer coe∆cients A; B and assume that forms a canonical number
system, i.e. each x 2 Z[ ] admits a representation of the shape
x = a 0 + a 1 +    + a h h
with a i 2 f0; 1; : : : ; jBj 1g. 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 nite 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 di erent
approaches: Geometry of numbers and nite automata theory. Each of these
approaches has its advantages and allows to emphasize on di erent properties

  

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

 

Collections: Mathematics