Amorphic complexity of group actions with applications to quasicrystals

Abstract

In this article, we define amorphic complexity for actions of locally compact $\sigma$-compact amenable groups on compact metric spaces. Amorphic complexity, originally introduced for $\mathbb{Z}$-actions, is a topological invariant which measures the complexity of dynamical systems in the regime of zero entropy. We show that 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 Meyer's cut and project method. We provide sharp upper bounds on amorphic complexity of such systems. In doing so, we observe an intimate relationship between amorphic complexity and fractal geometry.

26 pages, AAM version

References in corpus (6)

• Pure Point Dynamical and Diffraction Spectra
• Examples in the entropy theory of countable group actions
• The structure of tame minimal dynamical systems for general groups
• Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies
• Quasi-uniform Convergence in Dynamical Systems Generated by an Amenable Group Action
• On tameness of almost automorphic dynamical systems for general groups

Cited by in corpus (1)

• Orbit separation dimension as complexity measure for primitive inflation tilings