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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