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

TL;DR

This paper computes the Ellis enveloping semigroup for specific point patterns in Euclidean space, providing explicit descriptions of its algebraic and topological structure, with applications to tiling vertex patterns.

Contribution

It explicitly describes the Ellis semigroup for almost canonical model sets and applies this to the Amman-Beenker tiling vertex pattern.

Findings

Explicit algebraic and topological description of the Ellis semigroup

Application to the vertex pattern of the Amman-Beenker tiling

Enhanced understanding of dynamical systems associated with model sets

Abstract

We consider certain point patterns of an Euclidean space and calculate the Ellis enveloping semigroup of their associated dynamical systems. The algebraic structure and the topology of the Ellis semigroup, as well as its action on the underlying space, are explicitly described. As an example, we treat the vertex pattern of the Amman-Beenker tiling of the plane.