### Abstract

We study operators defined on a Hilbert space defined by a self-affine Delone set Λ and show that the usual trace of a restriction of the operator to finite-dimensional subspaces satisfies a certain lim sup law controlled by traces on a certain subalgebra. The asymptotic traces are defined through asymptotic cycles, or R^{d}-invariant distributions of a dynamical system defined by Λ. We use this to refine Shubin’s trace formula for certain self-adjoint operators acting on ℓ^{2}(Λ) and show that the errors of convergence in Shubin’s formula are given by these traces.

Original language | English (US) |
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Pages (from-to) | 2575-2597 |

Number of pages | 23 |

Journal | Annales Henri Poincare |

Volume | 19 |

Issue number | 9 |

DOIs | |

State | Published - Sep 1 2018 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics

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## Cite this

Schmieding, S. E., & Treviño, R. (2018). Traces of Random Operators Associated with Self-Affine Delone Sets and Shubin’s Formula.

*Annales Henri Poincare*,*19*(9), 2575-2597. https://doi.org/10.1007/s00023-018-0700-8