Abstract
Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam, and Skau conjectured that all minimal, free \(\mathbb {Z}^d\) actions on Cantor sets admit “small cocycles.” These represent classes in \(H^1\) that are mapped to small vectors in \(\mathbb {R}^d\) by the Ruelle–Sullivan (RS) map. We show that there exist \(\mathbb {Z}^2\) actions where no such small cocycles exist, and where the image of \(H^1\) under RS is \(\mathbb {Z}^2\). Our methods involve tiling spaces and shape deformations, and along the way we prove a relation between the image of RS and the set of “virtual eigenvalues,” i.e., elements of \(\mathbb {R}^d\) that become topological eigenvalues of the tiling flow after an arbitrarily small change in the shapes and sizes of the tiles.
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Communicated by Jean Bellissard.
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Clark, A., Sadun, L. Small Cocycles, Fine Torus Fibrations, and a \(\varvec{\mathbb {Z}^{2}}\) Subshift with Neither. Ann. Henri Poincaré 18, 2301–2326 (2017). https://doi.org/10.1007/s00023-017-0579-9
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DOI: https://doi.org/10.1007/s00023-017-0579-9