Small Cocycles, Fine Torus Fibrations, and a Z²Subshift with Neither

Abstract

© 2017, The Author(s). Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam, and Skau conjectured that all minimal, free Zdactions on Cantor sets admit “small cocycles.” These represent classes in H1that are mapped to small vectors in Rdby the Ruelle–Sullivan (RS) map. We show that there exist Z2actions where no such small cocycles exist, and where the image of H1under RS is Z2. 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 Rdthat become topological eigenvalues of the tiling flow after an arbitrarily small change in the shapes and sizes of the tiles.