Johannes Kellendonk - Semigroup methods in aperiodic order theory

Abstract

One of the basic tools to study Aperiodic tiling is the theory of ergodic dynamical systems. Properties of tiling dynamical systems reflect properties of the tiling, a point of view which has been very fruitful in the diffraction theory of aperiodic media and also in the description of their topological phases.
We consider a tiling to be highly ordered if its associated dynamical system is “nearly equicontinuous”. By this we mean that the tiling dynamical system factors onto a group rotation so that the fibres of the factor map are finite.
The perhaps simplest class of such tilings is defined by constant shape substitutions on a finite alphabet A. In this case the substitution is given by a shape with a collection of maps from A to A. These maps define a semigroup whose algebraic structure can be employed to study factors of the dynamical system of various types. This, in turn, is also useful to study another semigroup associated to the dynamical system, namely the enveloping (or Ellis) semigroup.

Talk

An aperiodic order is in relation to tilings in an end space.

As a simple example, we look at a tilings of cubes. We will label the cubes with letter in an aperiodic way. (Decorations of tilings by cubes). The decoration is realised by a function ${\mathbb{Z}}^d\to \mathcal{A}$ (into an alphabet.)

Finite local complexity: Up to translation for any size there are only finitely many patches. Repetitivity: Given any patch, there is a radius such that each ball of radius $R$ contains a translated copy of the patch. Aperiodic tiling: If the tiling shifted by a $t\in {\mathbb{R}}^d$ is the same then $t$ must be $0$.

Usually, we do not look at one tiling along. For a given tiling, we look at all tilings ${\Omega }_{\mathcal{T}}$ which look locally like $\mathcal{T}$. Meaning they agree on some finite patches.

${\Omega }_{\mathcal{T}}$ carries a metric.. $T,T^{\prime }$ are $ϵ$-close, if they agree in the ball $B_{\frac{1}{ϵ}}$ “up to an error of $ϵ$“. So ${\Omega }_{\mathcal{T}}$ is a topological space. If $T\in {\Omega }_{\mathcal{T}}$, then also $T-t\in {\Omega }_{\mathcal{T}}$ ($T$ shifted by $t$), so we have a continuous action of $\alpha \in {\mathbb{R}}^d$ on ${\Omega }_{\mathcal{T}}$. $({\Omega }_{\mathcal{T}},\alpha )$ is the tiling dyn system assigned to $\mathcal{T}$.

Next: Choose a puncture in each tile, s.t. tiles which are translated agree an the puncture. $X_{\mathcal{T}}=\{T\in |{\Omega }_{\mathcal{T}}\text{ a puncture of T lies on 0}\}$

Returning to the map, that assigns labels to the cubes. $X_{\mathcal{T}}$ can be understood as a subset of the maps ${\mathbb{Z}}^d\to \mathcal{A}$ (i.e. we shift the labels). There are many options how $\mathcal{T}$ might look like.

1. It is fully periodic.
2. It has Finite local complexity and is repetetive
3. it is a Math box manifold

Why is all of this stuff interesting?

1. It describes defraction of aperiodic materials (quasicrystals) A dynamical comes with a spectrum. For this, we take the Ergodic measure, we look at an $L^2$ function set, define an action on it and calculate the spectrum of the unitory representation of this action.
2. Something with $C^x$-algebra. There is a product structure which is the basic ingredient to the topological properties of aperiodic materials.
3. TDS? are nice complicated DS which nevertheless are tractable

Keywords

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