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Let T = T(A, D) be a self-affine attractor in
defined by an integral expanding matrix A and
a digit set D. In the
first part of this paper, in connection with canonical number systems,
we study connectedness of T when
D corresponds to the set of
consecutive integers
. It is shown that in
and
, for any integral expanding matrix A, T(A, D) is connected.
In the second part, we study connectedness of Pisot dual tiles, which
play an important role in the study of
-expansions, substitutions and
symbolic dynamical systems. It is shown that each tile of the dual
tiling 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. We even give a simple
necessary and sufficient condition of connectedness of the Pisot dual
tiles of degree 4. Detailed proofs will be given in [4].
Received: 2 March 2003 相似文献
2.
Periodica Mathematica Hungarica - 相似文献
3.
We study the connectedness of Pisot dual tilings. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4 which have infinitely many connected components. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, we give a complete classification of the β expansion of 1 for quartic Pisot units. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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