389 research outputs found
Dynamics of Irreducible Endomorphisms of
We consider the class non-surjective irreducible endomorphisms of the free
group . We show that such an endomorphism is topologically
represented by a simplicial immersion of a marked graph
; along the way we classify the dynamics of acting on
: there are at most fixed points, all of which are
attracting. After imposing a necessary additional hypothesis on , we
consider the action of on the closure of the
Culler-Vogtmann Outer space. We show that acts on with
"sink" dynamics: there is a unique fixed point , which is
attracting; for any compact neighborhood of , there is
, such that . The proof uses certian
projections of trees coming from invariant length measures. These ideas are
extended to show how to decompose a tree in the boundary of Outer space by
considering the space of invariant length measures on ; this gives a
decomposition that generalizes the decomposition of geometric trees coming from
Imanishi's theorem.Comment: v3, 46 pages, corrected gap in decomposition resul
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