On dynamical properties of quasicrystals

In the first half of this thesis we study the properties of the dynamical hull associated with model sets arising from irregular Euclidean Cut and Project Schemes. We provide deterministic as well as probabilistic constructions of irregular windows whose associated Cut and Project Schemes yield Delone dynamical systems with positive topological entropy. Moreover, we provide a construction of an irregular window whose associated dynamical hull has zero topological entropy but admits a unique ergodic measure. Furthermore, we show that dynamical hulls of irregular model sets always admit an infinite independence set. Hence, the dynamics cannot be tame. We extend this proof to a more general setting and show that tame implies regular for almost automorphic group actions on compact spaces. In the second half of this thesis, we provide and discuss a generalization of the Cut and Project formalism. We show that this new formalism yields all sets generated by Euclidean Cut and Project Schemes as well as non-Meyer sets. Furthermore, we give a sufficient criterion to obtain Meyer sets by this formalism in Euclidean space.



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