  We prove that the space of actions of Z^d by C^1 (orientation-preserving)
diffeomorphisms of either the interval or the circle is connected by arcs. This
is proved by showing that all such actions can be C^0 conjugated via a
1-parameter family into diffeomorphisms that converge to either the trivial
action or an action by Euclidean rotations. Extensions for nilpotent group
actions are provided.
