Small cocycles, fine torus fibrations, and a ${\mathbb Z}^2$ subshift with neither

TL;DR

This paper constructs examples of minimal free ${ m Z}^d$ actions on Cantor sets where no small cocycles exist, challenging previous conjectures and linking the Ruelle-Sullivan map to virtual eigenvalues.

Contribution

It provides explicit examples of ${ m Z}^d$ actions lacking small cocycles and establishes a relation between the Ruelle-Sullivan map and virtual eigenvalues.

Findings

Existence of ${ m Z}^d$ actions without small cocycles.

The image of $H^1$ under RS can be ${ m Z}^d$.

Connection between the RS map and virtual eigenvalues.

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 ${\mathbb Z}^d$ actions where no such small cocycles exist, and where the image of $H^1$ under RS is ${\mathbb Z}^d$. 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.