An ergodic theorem for Delone dynamical systems and existence of the integrated density of states

TL;DR

This paper establishes an ergodic theorem for Delone dynamical systems and demonstrates the existence of the integrated density of states for associated random operators, advancing understanding in mathematical physics and dynamical systems.

Contribution

It introduces an ergodic theorem for Banach space valued functions on Delone systems and proves the uniform convergence of the integrated density of states.

Findings

Proved an ergodic theorem for Delone dynamical systems.

Established the existence of the integrated density of states.

Demonstrated uniform convergence in distribution for associated random operators.

Abstract

We study strictly ergodic Delone dynamical systems and prove an ergodic theorem for Banach space valued functions on the associated set of pattern classes. As an application, we prove existence of the integrated density of states in the sense of uniform convergence in distribution for the associated random operators.