Gap Labelling for Schr\"odinger Operators on Quasiperiodic Tilings

TL;DR

This paper extends gap labelling results to a broad class of quasiperiodic tilings, including Penrose tilings, showing their algebraic structure relates to crossed products and gap labels are determined by invariant measures.

Contribution

It demonstrates that for many quasiperiodic tilings, their algebra is stably isomorphic to a crossed product with ^d, enlarging the class of tilings with known gap label sets.

Findings

Algebra of these tilings is stably isomorphic to a crossed product with ^d.

Gap labels are determined by invariant measures on the hull.

Includes Penrose tilings within this class.

Abstract

For a large class of tilings, including those which are obtained by the generalized dual method from regular grids, it is shown that their algebra is stably isomorphic to a crossed product with $\Z^d$. Penrose tilings belong to this class. This enlarges the class of tilings of which can be shown that a set of possible gap labels is completely determined by an invariant measure on the hull.