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

TL;DR

This paper proves the Pruning Front Conjecture for a specific open set of Hénon parameters, showing their dynamics are subshifts of the two-sided two-shift, thus advancing understanding of complex dynamical systems.

Contribution

It establishes the conjecture for an open set of Hénon parameters far from unimodal, demonstrating these maps are prunings of the horseshoe with subshift dynamics.

Findings

Proves the conjecture for an open set of parameters

Shows the dynamics are subshifts of the two-sided two-shift

Identifies the parameter set as consisting of two connected components

Abstract

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.