105 research outputs found
The Local Structure of Tilings and their Integer Group of Coinvariants
The local structure of a tiling is described in terms of a multiplicative
structure on its pattern classes. The groupoid associated to the tiling is
derived from this structure and its integer group of coinvariants is defined.
This group furnishes part of the -group of the groupoid -algebra for
tilings which reduce to decorations of . The group itself as well as the
image of its state is computed for substitution tilings in case the
substitution is locally invertible and primitive. This yields in particular the
set of possible gap labels predicted by -theory for Schr\"odinger operators
describing the particle motion in such a tiling.Comment: 45 pages including 9 figures, LaTe
Cyclic cohomology for graded -algebras and its pairings with van Daele -theory
We consider cycles for graded -algebras (Real -algebras)
which are compatible with the -structure and the real structure. Their
characters are cyclic cocycles. We define a Connes type pairing between such
characters and elements of the van Daele -groups of the -algebra
and its real subalgebra. This pairing vanishes on elements of finite order. We
define a second type of pairing between characters and -group elements which
is derived from a unital inclusion of -algebras. It is potentially
non-trivial on elements of order two and torsion valued. Such torsion valued
pairings yield topological invariants for insulators. The two-dimensional
Kane-Mele and the three-dimensional Fu-Kane-Mele strong invariant are special
cases of torsion valued pairings. We compute the pairings for a simple class of
periodic models and establish structural results for two dimensional aperiodic
models with odd time reversal invariance.Comment: 57 page
Spectral triples from stationary Bratteli diagrams
We construct spectral triples for path spaces of stationary Bratteli diagrams
and study their associated mathematical objects, in particular their zeta
function, their heat kernel expansion and their Dirichlet forms. One of the
main difficulties to properly define a Dirichlet form concerns its domain. We
address this question in particular in the context of Pisot substitution tiling
spaces for which we find two types of Dirichlet forms: one of transversal type,
and one of longitudinal type. Here the eigenfunctions under the translation
action can serve as a good core for a non-trivial Dirichlet form. We find that
the infinitesimal generators can be interpreted as elliptic differential
operators on the maximal equicontinuous factor of the tiling dynamical system.Comment: Version 2, 49 page
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