Nilpotent dynamics in dimension one: Structure and smoothness

Kiran Parkhe*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


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 {Ii,j , Mi,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 Ii,j is trivial, while the restriction of the action to any Mi,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 C1+α action for any α < 1/d(G), where d(G) is the degree of polynomial growth of G.

Original languageEnglish (US)
Pages (from-to)2258-2272
Number of pages15
JournalErgodic Theory and Dynamical Systems
Issue number7
StatePublished - Oct 1 2016

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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