Irregular model sets and tame dynamics

Fuhrmann, G.; Glasner, E.; Jäger, T.; Oertel, C.

doi:10.1090/tran/8349

Irregular model sets and tame dynamics
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by G. Fuhrmann, E. Glasner, T. Jäger and C. Oertel PDF

Trans. Amer. Math. Soc. 374 (2021), 3703-3734 Request permission

Abstract:

We study the dynamical properties of irregular model sets and show that the translation action on their hull always admits an infinite independence set. The dynamics can therefore not be tame and the topological sequence entropy is strictly positive. Extending the proof to a more general setting, we further obtain that tame implies regular for almost automorphic group actions on compact spaces.

In the converse direction, we show that even in the restrictive case of Euclidean cut and project schemes irregular model sets may be uniquely ergodic and have zero topological entropy. This provides negative answers to questions by Schlottmann and Moody in the Euclidean setting.

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