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Title: Small Cocycles, Fine Torus Fibrations, and a Z^2 Subshift with Neither
Authors: Clark, Alex
Sadun, Lorenzo
First Published: 10-Apr-2017
Publisher: Springer Verlag
Citation: Annales Henri Poincaré, 2017, doi:10.1007/s00023-017-0579-9
Abstract: Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam, and Skau conjectured that all minimal, free ZdZd actions on Cantor sets admit “small cocycles.” These represent classes in H1H1 that are mapped to small vectors in RdRd by the Ruelle–Sullivan (RS) map. We show that there exist Z2Z2 actions where no such small cocycles exist, and where the image of H1H1 under RS is Z2Z2 . 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 RdRd that become topological eigenvalues of the tiling flow after an arbitrarily small change in the shapes and sizes of the tiles.
DOI Link: 10.1007/s00023-017-0579-9
ISSN: 1424-0637
eISSN: 1424-0661
Version: Publisher Version
Status: Peer-reviewed
Type: Journal Article
Rights: Copyright © the authors, 2017. This is an open-access article distributed under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Appears in Collections:Published Articles, Dept. of Mathematics

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