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

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.