13 research outputs found
A characterization of subshifts with bounded powers
We consider minimal, aperiodic symbolic subshifts and show how to
characterize the combinatorial property of bounded powers by means of a metric
property. For this purpose we construct a family of graphs which all
approximate the subshift space, and define a metric on each graph which extends
to a metric on the subshift space. The characterization of bounded powers is
then given by the Lipschitz equivalence of a suitably defined infimum metric
with the corresponding supremum metric. We also introduce zeta-functions and
relate their abscissa of convergence to various exponents of complexity of the
subshift.Comment: 20 pages, 1 figur
Tiling groupoids and Bratteli diagrams
Let T be an aperiodic and repetitive tiling of R^d with finite local
complexity. Let O be its tiling space with canonical transversal X. The tiling
equivalence relation R_X is the set of pairs of tilings in X which are
translates of each others, with a certain (etale) topology. In this paper R_X
is reconstructed as a generalized "tail equivalence" on a Bratteli diagram,
with its standard AF-relation as a subequivalence relation.
Using a generalization of the Anderson-Putnam complex, O is identified with
the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram B is
built from this sequence, and its set of infinite paths dB is homeomorphic to
X. The diagram B is endowed with a horizontal structure: additional edges that
encode the adjacencies of patches in T. This allows to define an etale
equivalence relation R_B on dB which is homeomorphic to R_X, and contains the
AF-relation of "tail equivalence".Comment: 34 pages, 4 figure
PV cohomology of pinwheel tilings, their integer group of coinvariants and gap-labelling
In this paper, we first remind how we can see the "hull" of the pinwheel
tiling as an inverse limit of simplicial complexes (Anderson and Putnam) and we
then adapt the PV cohomology introduced in a paper of Bellissard and Savinien
to define it for pinwheel tilings. We then prove that this cohomology is
isomorphic to the integer \v{C}ech cohomology of the quotient of the hull by
which let us prove that the top integer \v{C}ech cohomology of the hull
is in fact the integer group of coinvariants on some transversal of the hull.
The gap-labelling for pinwheel tilings is then proved and we end this article
by an explicit computation of this gap-labelling, showing that \mu^t
\big(C(\Xi,\ZZ) \big) = \dfrac{1}{264} \ZZ [\dfrac{1}{5}].Comment: Problems of compilation by arxiv for figures on p.6 and p.7. I have
only changed these figure