426 research outputs found
Self-Similar Tilings of Fractal Blow-Ups
New tilings of certain subsets of are studied, tilings
associated with fractal blow-ups of certain similitude iterated function
systems (IFS). For each such IFS with attractor satisfying the open set
condition, our construction produces a usually infinite family of tilings that
satisfy the following properties: (1) the prototile set is finite; (2) the
tilings are repetitive (quasiperiodic); (3) each family contains
self-similartilings, usually infinitely many; and (4) when the IFS is rigid in
an appropriate sense, the tiling has no non-trivial symmetry; in particular the
tiling is non-periodic
Minkowski measurability results for self-similar tilings and fractals with monophase generators
In a previous paper [arXiv:1006.3807], the authors obtained tube formulas for
certain fractals under rather general conditions. Based on these formulas, we
give here a characterization of Minkowski measurability of a certain class of
self-similar tilings and self-similar sets. Under appropriate hypotheses,
self-similar tilings with simple generators (more precisely, monophase
generators) are shown to be Minkowski measurable if and only if the associated
scaling zeta function is of nonlattice type. Under a natural geometric
condition on the tiling, the result is transferred to the associated
self-similar set (i.e., the fractal itself). Also, the latter is shown to be
Minkowski measurable if and only if the associated scaling zeta function is of
nonlattice type.Comment: 18 pages, 1 figur
Fractal spectral triples on Kellendonk's -algebra of a substitution tiling
We introduce a new class of noncommutative spectral triples on Kellendonk's
-algebra associated with a nonperiodic substitution tiling. These spectral
triples are constructed from fractal trees on tilings, which define a geodesic
distance between any two tiles in the tiling. Since fractals typically have
infinite Euclidean length, the geodesic distance is defined using
Perron-Frobenius theory, and is self-similar with scaling factor given by the
Perron-Frobenius eigenvalue. We show that each spectral triple is
-summable, and respects the hierarchy of the substitution system. To
elucidate our results, we construct a fractal tree on the Penrose tiling, and
explicitly show how it gives rise to a collection of spectral triples.Comment: Updated to agree with published versio
Algorithm for determining pure pointedness of self-affine tilings
Overlap coincidence in a self-affine tiling in is equivalent to pure
point dynamical spectrum of the tiling dynamical system. We interpret the
overlap coincidence in the setting of substitution Delone set in and
find an efficient algorithm to check the pure point dynamical spectrum. This
algorithm is easy to implement into a computer program. We give the program and
apply it to several examples. In the course the proof of the algorithm, we show
a variant of the conjecture of Urba\'nski (Solomyak \cite{Solomyak:08}) on the
Hausdorff dimension of the boundaries of fractal tiles.Comment: 21 pages, 3 figure
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