Infinite powers and Cohen reals

We give a consistent example of a zero-dimensional separable metrizable space $Z$ such that every homeomorphism of $Z^{\omega }$ acts like a permutation of the coordinates almost everywhere. Furthermore, this permutation varies continuously. This shows that a result of Dow and Pearl is sharp, and gives some insight into an open problem of Terada. Our example $Z$ is simply the set of ${\omega }_1$ Cohen reals, viewed as a subspace of $2^{\omega }$.