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On the connectedness of self-affine attractors Shigeki AKIYAMA
 

Summary: On the connectedness of self-affine attractors
Shigeki AKIYAMA
and Nertila GJINI
Abstract
Let T = T(A, D) be a self-affine attractor in Rn
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 {0, 1, . . . , | det(A)| - 1}. It
is shown that in R3
and R4
, 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

  

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

 

Collections: Mathematics