44 research outputs found
On the noncommutative geometry of tilings
This is a chapter in an incoming book on aperiodic order. We review results
about the topology, the dynamics, and the combinatorics of aperiodically
ordered tilings obtained with the tools of noncommutative geometry
Regularity of aperiodic minimal subshifts
At the turn of this century Durand, and Lagarias and Pleasants established
that key features of minimal subshifts (and their higher-dimensional analogues)
to be studied are linearly repetitive, repulsive and power free. Since then,
generalisations and extensions of these features, namely -repetitive,
-repulsive and -finite (), have been introduced
and studied. We establish the equivalence of -repulsive and
-finite for general subshifts over finite alphabets. Further, we
studied a family of aperiodic minimal subshifts stemming from Grigorchuk's
infinite -group . In particular, we show that these subshifts provide
examples that demonstrate -repulsive (and hence -finite) is not
equivalent to -repetitive, for . We also give necessary and
sufficient conditions for these subshifts to be -repetitive, and
-repulsive (and hence -finite). Moreover, we obtain an explicit
formula for their complexity functions from which we deduce that they are
uniquely ergodic.Comment: 15 page
Spectral triples for subshifts
We propose a construction for spectral triple on algebras associated with
subshifts. One-dimensional subshifts provide concrete examples Z-actions on
Cantor sets. The C*-algebra of this dynamical system is generated by functions
in C(X) and a unitary element u implementing the action. Building on ideas of
Christensen and Ivan, we give a construction of a family of spectral triples on
the commutative algebra C(X). There is a canonical choice of eigenvalues for
the Dirac operator D which ensures that [D,u] is bounded, so that it extends to
a spectral triple on the crossed product.
We study the summability of this spectral triple, and provide examples for
which the Connes' distance associated with it on the commutative algebra is
unbounded, and some for which it is bounded. We conjecture that our results on
the Connes distance extend to the spectral triple defined on the noncommutative
algebra.Comment: The exposition was shortened at some places and clarified at others.
The proof of Theorem 5.8 had a slight gap which we fixed (without consequence
for the result
Around groups in Hilbert Geometry
This is survey about action of group on Hilbert geometry. It will be a
chapter of the "Handbook of Hilbert geometry" edited by G. Besson, M. Troyanov
and A. Papadopoulos.Comment: ~60 page
Emergence of Structures in Particle Systems: Mechanics, Analysis and Computation
The meeting focused on the last advances in particle systems. The talks covered a broad range of topics, ranging from questions in crystallization and atomistic systems to mesoscopic models of defects to machine learning approaches and computational aspects
Diffraction of stochastic point sets : exactly solvable examples
Stochastic point sets are considered that display a diffraction
spectrum of mixed type, with special emphasis on explicitly computable cases
together with a unified approach of reasonable generality. Several pairs of
autocorrelation and diffraction measures are discussed that show a duality
structure that may be viewed as analogues of the Poisson summation formula
for lattice Dirac combs