Let
$M$ be a connected
$1$-manifold, and let
$G$ be a finitely-generated nilpotent group of homeomorphisms of
$M$. Our main result is that one can find a collection
$\{I_{i,j},M_{i,j}\}$ of open disjoint intervals with dense union in
$M$, such that the intervals are permuted by the action of
$G$, and the restriction of the action to any
$I_{i,j}$ is trivial, while the restriction of the action to any
$M_{i,j}$ is minimal and abelian. It is a classical result that if
$G$ is a finitely-generated, torsion-free nilpotent group, then there exist faithful continuous actions of
$G$ on
$M$. Farb and Franks [Groups of homeomorphisms of one-manifolds, III: Nilpotent subgroups. Ergod. Th. & Dynam. Sys.23 (2003), 1467–1484] showed that for such
$G$, there always exists a faithful
$C^{1}$ action on
$M$. As an application of our main result, we show that every continuous action of
$G$ on
$M$ can be conjugated to a
$C^{1+\unicode[STIX]{x1D6FC}}$ action for any
$\unicode[STIX]{x1D6FC}<1/d(G)$, where
$d(G)$ is the degree of polynomial growth of
$G$.