TY - JOUR

T1 - Nilpotent dynamics in dimension one

T2 - Structure and smoothness

AU - Parkhe, Kiran

N1 - Publisher Copyright:
© Cambridge University Press, 2015.

PY - 2016/10/1

Y1 - 2016/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84931003267&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84931003267&partnerID=8YFLogxK

U2 - 10.1017/etds.2015.8

DO - 10.1017/etds.2015.8

M3 - Article

AN - SCOPUS:84931003267

VL - 36

SP - 2258

EP - 2272

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 7

ER -