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 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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References
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)
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)
Bellissard, J.: Gap Labelling Theorems for Schrödinger Operators. From number theory to physics (Les Houches,1989). Springer, Berlin, pp. 538–630 (1992)
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)
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)
Einsiedler, M., Ward, T.: Ergodic Theory with a View Towards Number Theory. Graduate Texts in Mathematics, vol. 259. Springer, London (2011)
Kellendonk, J.: Noncommutative geometry of tilings and gap labelling. Rev. Math. Phys. 7(7), 1133–1180 (1995)
Kellendonk, J.: Pattern-equivariant functions and cohomology. J. Phys. A 36(21), 5765–5772 (2003)
Kellendonk, J., Putnam, I.F.: The Ruelle–Sullivan map for actions of \({\mathbb{R}}^n\). Math. Ann. 334(3), 693–711 (2006)
Lee, J.-Y., Moody, R.V., Solomyak, B.: Pure point dynamical and diffraction spectra. Ann. Henri Poincaré 3(5), 1003–1018 (2002)
Lenz, D., Stollmann, P.: Algebras of random operators associated to Delone dynamical systems. Math. Phys. Anal. Geom. 6(3), 269–290 (2003)
Sadun, L.: Topology of Tiling Spaces. University Lecture Series, vol. 46. American Mathematical Society, Providence (2008)
Schmieding, S., Treviño, R.: Self affine Delone sets and deviation phenomena. Commun. Math. Phys. 357(3), 1071–1112 (2018)
Acknowledgements
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.
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Communicated by Stéphane Nonnenmacher.
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Schmieding, S., Treviño, R. Traces of Random Operators Associated with Self-Affine Delone Sets and Shubin’s Formula. Ann. Henri Poincaré 19, 2575–2597 (2018). https://doi.org/10.1007/s00023-018-0700-8
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DOI: https://doi.org/10.1007/s00023-018-0700-8