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Nilpotent dynamics in dimension one: structure and smoothness

Published online by Cambridge University Press:  15 June 2015

KIRAN PARKHE
KIRAN PARKHE*
Affiliation:
Faculty of Mathematics, Technion – Israel Institute of Technology, Technion City, Haifa 32000, Israel email [email protected]
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Abstract

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 $G$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 7 , October 2016 , pp. 2258 - 2272
Copyright
© Cambridge University Press, 2015 

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