Amorphic complexity of group actions with applications to quasicrystals

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 ze...

Full description

Saved in:
Bibliographic Details
Published in:Transactions of the American Mathematical Society 2023-04, Vol.376 (4), p.2395
Main Authors: Fuhrmann, Gabriel, Gröger, Maik, Jäger, Tobias, Kwietniak, Dominik
Format: Article
Language:English
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/8700