The giant spiral (Q1279628)

Cocycles of \(\mathbb{Z}^m\) actions on compact metric spaces provide a means for constructing \(\mathbb{R}^m\) actions or flows, called suspension flows. It is known that all \(\mathbb{R}^m\) flows with a free dense orbit have an almost one-to-one extension which is a suspension flow. When \(m=1\), examples of cocycles are easy to construct; there is a one-to-one correspondence between cocycles and real valued continuous functions. However, when \(m>1\), the construction of examples of cocycles becomes more problematic. The only existing class of examples, the close to linear cocycles, have strong linearity properties and are well understood. In fact, when the \(\mathbb{Z}^m\) action is uniquely ergodic, all cocycles are close to linear. We show here that in general this need not be the case. We present a method, suggested to us by Hillel Furstenberg, for constructing examples of cocycles when \(m>1\) and use this method to construct a non close to linear cocycle on a minimal \(\mathbb{Z}^2\) action.