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Fixed points of nilpotent actions on $\mathbb{S}^{2}$

Published online by Cambridge University Press:  05 August 2014

JAVIER RIBÓN
JAVIER RIBÓN*
Affiliation:
Instituto de Matemática, UFF, Rua Mário Santos Braga S/N Valonguinho, Niterói, Rio de Janeiro, 24020-140, Brasil email javier@mat.uff.br
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Abstract

We prove that a nilpotent subgroup of orientation-preserving $C^{1}$ diffeomorphisms of $\mathbb{S}^{2}$ has a finite orbit of cardinality at most two. We also prove that a finitely generated nilpotent subgroup of orientation-preserving $C^{1}$ diffeomorphisms of $\mathbb{R}^{2}$ preserving a compact set has a global fixed point. These results generalize theorems of Franks et al for the abelian case. We show that a nilpotent subgroup of orientation-preserving $C^{1}$ diffeomorphisms of $\mathbb{S}^{2}$ that has a finite orbit of odd cardinality also has a global fixed point. Moreover, we study the properties of the 2-points orbits of nilpotent fixed-point-free subgroups of orientation-preserving $C^{1}$ diffeomorphisms of $\mathbb{S}^{2}$.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 1 , February 2016 , pp. 173 - 197
Copyright
© Cambridge University Press, 2014 

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