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 C¹ 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+α action for any α < 1/d(G), where d(G) is the degree of polynomial growth of G.