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Rotation Numbers and Rotation Classes on One-Dimensional Tiling Spaces

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Abstract

We extend rotation theory of circle maps to tiling spaces. Specifically, we consider a one-dimensional tiling space \(\Omega \) with finite local complexity and study self-maps F that are homotopic to the identity and whose displacements are strongly pattern equivariant. In place of the familiar rotation number, we define a cohomology class \([\mu ] \in {\check{H}}^1(\Omega , {\mathbb {R}})\). We prove existence and uniqueness results for this class, develop a notion of irrationality, and prove an analogue of Poincaré’s theorem: If \([\mu ]\) is irrational, then F is semi-conjugate to uniform translation on a space \(\Omega _\mu \) of tilings that is homeomorphic to \(\Omega \). In such cases, F is semi-conjugate to uniform translation on \(\Omega \) itself if and only if \([\mu ]\) lies in a certain subspace of \({\check{H}}^1(\Omega , {\mathbb {R}})\).

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Notes

  1. The FLC condition is usually defined in terms of local patches, but in one dimension it is equivalent to simply having finitely many possible kinds of tiles.

  2. In the literature, “pattern equivariant” usually means sPE. The term “PE cohomology” was defined before the theory of wPE forms was developed.

  3. Note that \(\rho \) is a dimensionful quantity, having units of length. As such, the naive definition of \(\rho \) being an irrational number does not make sense.

  4. When f is \(C^2\) but \(f'\) is zero at isolated points, the argument is technically more complicated, but follows the same overall strategy.

  5. Applying the substitution \(\sigma \) recursively, one obtains a list of increasingly long words \(\sigma ^n(a)\) and \(\sigma ^n(b)\). The allowed bi-infinite words are those for which every finite sub-word is found in one of these n-times substituted letters.

  6. It is always possible to do a shape change to make all the tiles have integer length, so every tiling space is homeomorphic to a Cantor bundle over a circle. Moreover, every higher dimensional minimal FLC tiling space is homeomorphic to a Cantor bundle over a torus. [37].

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Acknowledgements

J.A.-P. acknowledges financial support from CONICYT FONDECYT REGULAR 1160975, CHILE. The authors thank Michael Baake, Alex Clark, Franz Gähler, Antoine Julien, Johannes Kellendonk, John Hunton, Jamie Walton, and Dan Rust for helpful discussions.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Andres Bello, Republica 498, Santiago, Chile

    José Aliste-Prieto

  2. Department of Mathematics, Texas Lutheran University, Seguin, TX, 78155, USA

    Betseygail Rand

  3. Department of Mathematics, University of Texas, Austin, TX, 78712, USA

    Lorenzo Sadun

Authors
  1. José Aliste-Prieto
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  2. Betseygail Rand
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  3. Lorenzo Sadun
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Correspondence to José Aliste-Prieto.

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Communicated by Dmitry Dolgopyat.

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Aliste-Prieto, J., Rand, B. & Sadun, L. Rotation Numbers and Rotation Classes on One-Dimensional Tiling Spaces. Ann. Henri Poincaré 22, 2161–2193 (2021). https://doi.org/10.1007/s00023-021-01019-2

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