Nilpotent dynamics in dimension one: structure and smoothness

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 $G$.