108 research outputs found
On the connectedness of planar self-affine sets
In this paper, we consider the connectedness of planar self-affine set
arising from an integral expanding matrix with
characteristic polynomial and a digit set
. The necessary and sufficient conditions only
depending on are given for the to be connected.
Moreover, we also consider the case that is non-consecutively
collinear.Comment: 18 pages; 18 figure
Connectedness of planar self-affine sets associated with non-consecutive collinear digit sets
In the paper, we focus on the connectedness of planar self-affine sets
generated by an integer expanding matrix with and a collinear digit set , where and
such that is linearly independent. We discuss
the domain of the digit to determine the connectedness of
. Especially, a complete characterization is obtained when
we restrict to be an integer. Some results on the general case of are obtained as well.Comment: 15 pages, 10 figure
Boundaries of Disk-like Self-affine Tiles
Let be a disk-like self-affine tile generated by an
integral expanding matrix and a consecutive collinear digit set , and let be the characteristic polynomial of . In the
paper, we identify the boundary with a sofic system by
constructing a neighbor graph and derive equivalent conditions for the pair
to be a number system. Moreover, by using the graph-directed
construction and a device of pseudo-norm , we find the generalized
Hausdorff dimension where
is the spectral radius of certain contact matrix . Especially,
when is a similarity, we obtain the standard Hausdorff dimension where is the largest positive zero of
the cubic polynomial , which is simpler than
the known result.Comment: 26 pages, 11 figure
Rational self-affine tiles
An integral self-affine tile is the solution of a set equation , where
is an integer matrix and is a finite
subset of . In the recent decades, these objects and the induced
tilings have been studied systematically. We extend this theory to matrices
. We define rational self-affine tiles
as compact subsets of the open subring of the ad\'ele ring , where the factors of the
(finite) product are certain -adic completions of a number field
that is defined in terms of the characteristic polynomial of .
Employing methods from classical algebraic number theory, Fourier analysis in
number fields, and results on zero sets of transfer operators, we establish a
general tiling theorem for these tiles. We also associate a second kind of
tiles with a rational matrix. These tiles are defined as the intersection of a
(translation of a) rational self-affine tile with . Although these intersection
tiles have a complicated structure and are no longer self-affine, we are able
to prove a tiling theorem for these tiles as well. For particular choices of
digit sets, intersection tiles are instances of tiles defined in terms of shift
radix systems and canonical number systems. Therefore, we gain new results for
tilings associated with numeration systems
- …