## Institute for Mathematical Sciences

## Preprint ims01-10

** A. de Carvalho and T. Hall**
* How to prune a horseshoe*

Abstract: Let $F\colon\ofr^2\to\ofr^2$ be a homeomorphism. An open $F$-invariant subset $U$ of $\ofr^2$ is a {\em pruning region}
for $F$ if it is possible to deform $F$ continuously to a
homeomorphism $F_U$ for which every point of $U$ is wandering,
but which has the same dynamics as $F$ outside of $U$. This
concept was motivated by the {\em Pruning Front Conjecture} of
Cvitanovi\'c, Gunaratne, and Procaccia, which claims that every
H\'enon map can be understood as a pruned horseshoe. This paper
is a survey of pruning theory, concentrating on prunings of the
horseshoe. We describe conditions on a disk $D$ which ensure
that the orbit of its interior is a pruning region; explain how
prunings of the horseshoe can be understood in terms of
underlying tree maps; discuss the connection between pruning and
Thurston's classification theorem for surface homeomorphisms;
motivate a conjecture describing the {\em forcing relation} on
horseshoe braid types; and use this theory to give a precise
statement of the pruning front conjecture.

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