On close to linear cocycles

Abstract:

If we have a flow $(X,\Bbb {Z}^m)$ and a cocycle $h$ on this flow, $h:X\times \Bbb {Z}^m\rightarrow \Bbb {R}^m$, then $h$ is called close to linear if $h$ can be written as the direct sum of a linear (constant) cocycle and a cocycle in the closure of the coboundaries. Many of the desirable consequences of linearity hold for such cocycles and, in fact, a close to linear cocycle is cohomologous to a cocycle which is norm close to a linear one. Furthermore in the uniquely ergodic case all cocycles are close to linear. We also establish that a close to linear cocycle which is covering is cohomologous to one with the special property that it can be extended by piecewise linearity to an invertible cocycle from $X\times \Bbb {R}^m$ to itself. This implies that a suspension obtained from a close to linear cocycle is isomorphic to a time change of the suspension obtained from the identity cocycle.