Annales Henri Poincaré

, Volume 19, Issue 9, pp 2575–2597 | Cite as

Traces of Random Operators Associated with Self-Affine Delone Sets and Shubin’s Formula

  • Scott Schmieding
  • Rodrigo TreviñoEmail author


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 certain self-adjoint operators acting on \(\ell ^2(\Lambda )\) and show that the errors of convergence in Shubin’s formula are given by these traces.


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We would like to thank J. Kellendonk for discussing questions related to Shubin’s formula and I. Putnam for discussing traces on \(*\)-algebras. Part of R.T.’s travel funding for this project came from a AMS-Simons Travel Grant.


  1. 1.
    Anderson, J.E., Putnam, I.F.: Topological invariants for substitution tilings and their associated \(C^*\)-algebras. Ergod. Theory Dyn. Syst. 18(3), 509–537 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bellissard, J.: \(K\)-theory of \(C^\ast \)-algebras in solid state physics. In: Statistical Mechanics and Field Theory: Mathematical Aspects (Groningen, 1985), Lecture Notes in Physics, vol. 257. Springer, Berlin, pp. 99–156 (1986)Google Scholar
  3. 3.
    Bellissard, J.: Gap Labelling Theorems for Schrödinger Operators. From number theory to physics (Les Houches,1989). Springer, Berlin, pp. 538–630 (1992)Google Scholar
  4. 4.
    Bellissard, J., Herrmann, D.J.L., Zarrouati, M.: Hulls of aperiodic solids and gap labeling theorems. In: Baake, M., Moody, R. (eds.) Directions in Mathematical Quasicrystals, CRM Monograph Series, vol. 13. American Mathematical Society, Providence, pp. 207–258 (2000)Google Scholar
  5. 5.
    Damanik, D., Embree, M., Gorodetski, A.: Spectral properties of Schrödinger operators arising in the study of quasicrystals. In: Mathematics of Aperiodic Order, Progress in Mathematical Physics, vol. 309. Birkhäuser, Basel, pp. 307–370 (2015)Google Scholar
  6. 6.
    Einsiedler, M., Ward, T.: Ergodic Theory with a View Towards Number Theory. Graduate Texts in Mathematics, vol. 259. Springer, London (2011)zbMATHGoogle Scholar
  7. 7.
    Kellendonk, J.: Noncommutative geometry of tilings and gap labelling. Rev. Math. Phys. 7(7), 1133–1180 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kellendonk, J.: Pattern-equivariant functions and cohomology. J. Phys. A 36(21), 5765–5772 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kellendonk, J., Putnam, I.F.: The Ruelle–Sullivan map for actions of \({\mathbb{R}}^n\). Math. Ann. 334(3), 693–711 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lee, J.-Y., Moody, R.V., Solomyak, B.: Pure point dynamical and diffraction spectra. Ann. Henri Poincaré 3(5), 1003–1018 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lenz, D., Stollmann, P.: Algebras of random operators associated to Delone dynamical systems. Math. Phys. Anal. Geom. 6(3), 269–290 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sadun, L.: Topology of Tiling Spaces. University Lecture Series, vol. 46. American Mathematical Society, Providence (2008)zbMATHGoogle Scholar
  13. 13.
    Schmieding, S., Treviño, R.: Self affine Delone sets and deviation phenomena. Commun. Math. Phys. 357(3), 1071–1112 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics DepartmentEvanstonUSA
  2. 2.Department of MathematicsCollege ParkUSA

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