Amorphic complexity of group actions with applications to quasicrystals

TL;DR

This paper introduces amorphic complexity for group actions on compact metric spaces, focusing on zero entropy systems like Delone dynamical systems, and explores its connections to fractal geometry.

Contribution

It extends amorphic complexity to actions of locally compact amenable groups and provides bounds for systems related to quasicrystals.

Findings

Amorphic complexity effectively measures complexity in zero entropy group actions.

Sharp upper bounds are established for Delone dynamical systems.

A link between amorphic complexity and fractal geometry is demonstrated.

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.