How to prune a horseshoe

Let F : ²→² be a homeomorphism. An open F-invariant subset U of ² is a 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 is motivated by the Pruning Front Conjecture (PFC) introduced by Cvitanovic, which claims that every Hénon map can be understood as a pruned horseshoe.

This paper contains recent results in 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 forcing relation on horseshoe braid types; and use this theory to give a precise statement of the PFC.