Traces of random operators associated with self-affine Delone sets and Shubin's formula

TL;DR

This paper investigates the asymptotic behavior of traces of operators on Hilbert spaces associated with self-affine Delone sets, refining Shubin's trace formula by linking convergence errors to invariant distributions of the underlying dynamical system.

Contribution

It introduces a new approach to analyze the asymptotic traces of operators on self-affine Delone sets, refining existing trace formulas with a detailed understanding of convergence errors.

Findings

Asymptotic traces are governed by invariant distributions of the dynamical system.

Refinement of Shubin's trace formula for self-adjoint operators.

Errors in the convergence of Shubin's formula are characterized by these traces.

Abstract

We study operators defined on a Hilbert space defined by a self-affine Delone set $\Lambda$ and show that the usual trace of a restriction of the operator to finite-dimensional subspaces satisfies a certain $\limsup$ law controlled by traces on a certain subalgebra. The asymptotic traces are defined through asymptotic cycles, or $\mathbb{R}^d$-invariant distributions of a dynamical system defined by $\Lambda$. We use this to refine Shubin's trace formula for self-adjoint operators and show that the errors of convergence in Shubin's formula are given by these traces.