Penrose Tilings, Chaotic Dynamical Systems and Algebraic K-Theory

TL;DR

This paper explores the complex structure of Penrose tilings and the Arnold cat map using noncommutative geometry, associating C*-algebras to better understand their phase space than traditional topology.

Contribution

It introduces a novel approach of using noncommutative C*-algebras and K-theory to analyze dynamical systems like Penrose tilings and the Arnold cat map, surpassing traditional methods.

Findings

K-groups of associated C*-algebras reveal detailed phase space structure

Noncommutative geometry provides a more accurate framework than topology

Method suggests broader applicability to other dynamical systems

Abstract

After investigating by examples the unusual and striking elementary properties of the Penrose tilings and the Arnold cat map, we associate a finite symbolic dynamics with finite grammar rules to each of them. Instead of studying these Markovian systems with the help of set-topology, which would give only pathological results, a noncommutative approximately finite C*-algebra is associated to both systems. By calculating the K-groups of these algebras it is demonstrated that this noncommutative point of view gives a much more appropriate description of the phase space structure of these systems than the usual topological approach. With these specific examples it is conjectured that the methods of noncommutative geometry could be successfully applied to a wider class of dynamical systems.