Abstract
Let T=T(A,D) be a self-affine tile in \reals^n defined by an integral expanding matrix A and a digit set D. In connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers \0,1,..., |det(A)|-1\. It is shown that in \reals^3 and \reals^4, for any integral expanding matrix A, T(A,D) is connected. We also study the connectedness of Pisot dual tilings which play an important role in the study of β -expansion, substitution and symbolic dynamical system. It is shown that each tile 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. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, a complete classification of the β -expansion of 1 for quartic Pisot units is given.
Highlights
A non empty set in Rn is called a tile (i) if it coincides with the closure of its interior
We will discuss the connectedness of tiles which arise from two different kinds of number systems
In [5], we introduced a wider class of Pisot units with this tiling property called weakly finiteness
Summary
A non empty set in Rn is called a tile (i) if it coincides with the closure of its interior. Even when p(0) = −1 there are many such cases Including these cases, we can generalize the above conjecture: Conjecture 1 Let β be a Pisot unit and consider its eventually periodic β-expansion of 1 : dβ(1) = .d−1, · · · , d−n, (d−n−1, · · · , d−n−k)ω. Theorem 4.3 on page 290 was proved by Bassino [11] She computed the β-expansion of 1 for any cubic Pisot number, including non units. We give a proof of Theorem 1.2 on page 273 and describe a method to prove connectedness of Pisot dual tiles.
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