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Journal of Modern Dynamics (JMD)
 

Fixed points of Abelian actions

Pages: 443 - 464, Issue 3, July 2007      doi:10.3934/jmd.2007.1.443

 
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John Franks - Department of Mathematics, Northwestern University, Evanston, Illinois, United States (email)
Michael Handel - Department of Mathematics and Computer Science, Herbert H. Lehman College (CUNY), New York, United States (email)
Kamlesh Parwani - Department of Mathematics, Eastern Illinois University, Illinois, United States (email)

Abstract: We prove that if $\mathfrak{F}$ is an abelian group of $C^1$ diffeomorphisms isotopic to the identity of a closed surface $S$ of genus at least two, then there is a common fixed point for all elements of $\mathfrak{F}$. If $\mathfrak{F}$ is an abelian group of $C^1$ diffeomorphisms (not necessarily isotopic to the identity) of a closed surface $S$ of genus at least two, then $\mathfrak{F}$ has a subgroup of finite index all of whose elements share a common fixed point.

Keywords:  Abelian group actions on surfaces, fixed points.
Mathematics Subject Classification:  Primary: 37E30, 57M60; Secondary: 57S25, 55M20.

Received: October 2006;      Revised: March 2007;      Available Online: April 2007.