From: yair@jericho.sunysb.edu (Yair Minsky)
Newsgroups: sci.math.research
Subject: Re: pseudo-Anosov homeomorphisms
Date: 30 Sep 1996 13:26:58 GMT
Summary: Yes
Keywords: Nielsen fixed-point
>
> Let S be a closed orientable surface of genus at least two, and
> let f:S -> S be a pseudo-Anosov homeomorphism. Must f have a
> fixed point?
>
> Jim
Hi Jim. The answer is yes. This is a consequence of Nielsen
fixed-point theory, which partitions the fixed points of a surface
homeomorphism into equivalence classes which then remain stable under
isotopy. The partition has to do with asymptotic behavior of orbits in
the universal cover. In particular, the standard representative of a
pseudo-Anosov isotopy class has the minimum possible number of fixed
points, one for each equivalence class.
There is a good survey of this theory by Phil Boyland, in
Boyland, Philip Topological methods in surface dynamics.
Topology Appl. 58 (1994), no. 3, 223--298.
Other names associated with this are Handel, and (naturally) Thurston,
but I don't really know the literature.
bye,
Yair Minsky