# '22

36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

### "Cantor actions with discrete spectrum and their amorphic complexity"

#### Gröger, Maik

Amorphic complexity, originally introduced for integer actions, is a topological invariant which measures the complexity of dynamical systems in the regime of zero entropy. We will introduce its definition for actions by locally compact $\sigma$-compact amenable groups on compact metric spaces. Further, we will illustrate some of its basic properties and show why it is tailor-made to study strictly ergodic group actions with discrete spectrum and continuous eigenfunctions. This class of actions includes, in particular, Delone dynamical systems related to regular model sets obtained via cut and project schemes (CPS). Moreover, there is a canonical way to associate such an action to locally non-degenerate (a notion recently introduced together with O. Lukina) equicontinuous Cantor actions via their holonomy. Finally, if time permits, we present some explicit bounds for amorphic complexity for actions related to CPS and inflation tilings.  This is joint work with G. Fuhrmann, T. Jäger & D. Kwietniak and O. Lukina.

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