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The Pruning Front Conjecture is proved for an open set of Hénon parameters far from unimodal. More specifically, for an open subset of Hénon parameter space, consisting of two connected components one of which intersects the area-preserving locus, it is shown that the associated Hénon maps are prunings of the horseshoe. In particular, their dynamics is a subshift of the two-sided two-shift. A different way to formalize pruning for the Lozi family was introduced by Ishii in [13] which enabled him to prove the PFC for that family.

In this paper, we present results for the forcing relation on the set of braid types of periodic orbits of a rotational horseshoe on the annulus. Precisely, we are concerned with a family of periodic orbits, called the Boyland family, and we prove that for each pair $(r,s)$ of rational numbers with $r<s$ in $(0,1)$, there exists a non-monotone orbit $B_{r,s}$ in this family which has pseudo-Anosov type and rotation interval $[r,s]$. Furthermore, the forcing relation among these orbits is given by the inclusion order on their rotation sets. It is also proved that the Markov partition associated to each Boyland orbit comes from a pruning map which projects to a bimodal circle map. This family also contains the Holmes orbits $H_{p/q}$, which are the largest for the forcing order among all the $(p,q)$-orbits of the rotational horseshoe.

In this paper we deal with the Boyland order of horseshoe orbits. We prove that there exists a set $\mathcal{R}$ of renormalizable horseshoe orbits containing only quasi-one-dimensional ones, that is, for these orbits the Boyland order coincides with the unimodal order