Proof of the Pruning Front Conjecture for certain H\'enon parameters

The Pruning Front Conjecture is proved for an open set of H\'enon parameters far from unimodal. More specifically, for an open subset of H\'enon parameter space, consisting of two connected components one of which intersects the area-preserving locus, it is shown that the associated H\'enon maps are prunings of the horseshoe. In particular, their dynamics is a subshift of the two-sided two-shift