Cohomology of Canonical Projection Tilings

Ellis enveloping semigroup for almost canonical model sets of an Euclidean space

“…However, our definition doesn't fit exactly with the one given in [16] (it can be shown that ours implies the one of Julien), but as we don't want to introduce another definition for something so close to the situation of [16], we allow ourselves to abuse terminology and call ours almost canonical. As shown in [8], a canonical window always satisfies Assumptions 1 and 2 and is thus almost canonical in our sense.…”

“…In this work we propose a qualitative description of the Ellis semigroup of dynamical systems associated with particular point patterns, the almost canonical model sets. These particular patterns are relevant in the crystallographic sense, as well as very accessible mathematically: One can get a complete picture of the hull X Λ 0 of such patterns (Le [19]), as well as their associated C * -Algebras (a recent source is Putnam [27], see references therein), and also compute their cohomology and K-theory groups (Forrest, Hunton and Kellendonk [8], Gähler, Hunton and Kellendonk [9] and Putnam [27]) as well as the asymptotic exponent of their complexity function (Julien [16]). We show that in our situation it is possible to completely describe elements of the Ellis semigroup, their action onto the underlying space, as well as the algebraic and topological structure of this semigroup.…”

Pattern-equivariant functions and cohomology

“…The topology of M P is then generated by U P,p,ǫ,y = {ω ∈ M P |∃x ∈ B ǫ (y) : P − p − x ∈ ω} (cf. [FHK02]). Moreover, we may restrict ǫ > 0 to values smaller than the minimal distance r 0 of two points in P. For such ǫ, U P,p,ǫ,y is homeomorphic to U P,p × B ǫ (y).…”

Bulk–Boundary Correspondence for Sturmian Kohmoto-Like Models

“…Here we deal exclusively with the one dimensional quasi-crystalline case and, as we shall see, the bulk-boundary correspondence is quite subtle. One of the basic results about the topology of quasi-crystals is that the phason degree of freedom lives in a Cantor set [34,25], which is a totally disconnected space. In contradistinction, for almost periodic systems, the phason lives on a circle.…”