  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.
