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Operators, Algebras and Their Invariants for Aperiodic Tilings

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Substitution and Tiling Dynamics: Introduction to Self-inducing Structures

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2273))

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Abstract

We review the construction of operators and algebras from tilings of Euclidean space. This is mainly motivated by physical questions, in particular after topological properties of materials. We explain how the physical notion of locality of interaction is related to the mathematical notion of pattern equivariance for tilings and how this leads naturally to the definition of tiling algebras. We give a brief introduction to the K-theory of tiling algebras and explain how the algebraic topology of K-theory gives rise to a correspondence between the topological invariants of the bulk and its boundary of a material.

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Notes

  1. 1.

    Originally the decorations encode matching conditions, but we will not make use of that here.

  2. 2.

    The Hausdorff metric between equivalence classes of compact sets is the infimum over the Hausdorff distances between representatives of the classes.

  3. 3.

    At low enough temperature.

  4. 4.

    In [37] one finds a formula for the same class with a different representative, the above is also be valid if A is a real C ∗-algebra.

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Authors and Affiliations

  1. Institute Camille Jordan, Université Claude Bernard Lyon1, Villeurbanne, France

    Johannes Kellendonk

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  1. Institute of Mathematics, University of Tsukuba, Tsukuba, Japan

    Shigeki Akiyama

  2. Institut de Mathématiques de Marseille (I2M), Aix-Marseille University, Marseille Cedex 09, France

    Pierre Arnoux

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Kellendonk, J. (2020). Operators, Algebras and Their Invariants for Aperiodic Tilings. In: Akiyama, S., Arnoux, P. (eds) Substitution and Tiling Dynamics: Introduction to Self-inducing Structures. Lecture Notes in Mathematics, vol 2273. Springer, Cham. https://doi.org/10.1007/978-3-030-57666-0_4

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