Proof of the Pruning Front Conjecture for certain Hénon parameters

Abstract

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.

Uncountably many planar embeddings of unimodal inverse limit spaces

“…One of the simplest examples is the Knaster continuum (bucket handle), which is the attractor of Smale's horseshoe map and can be modeled as the inverse limit space of the full tent map T 2 (x) := min{2x, 2(1 − x)} for x ∈ [0, 1], see [1]. The topological unfolding of the Smale's horseshoe has been a topic of ongoing interest, related to the pruning front conjecture for the Hénon family developed by Cvitanović et al in [10,11] and demonstrated more recently with the work of Boyland, de Carvalho & Hall [6] and Mendoza [17].…”

Kneading sequences for toy models of Hénon maps

“…This is known as the pruning front conjecture. Mendonza proved that the pruning front conjecture holds in an open set of the parameter space [10], and Ishii proved it for the Lozi family [7]. For more detail on the theory of pruning fronts and its relationship with kneading theory, see [2,3].…”

Kneading sequences for toy models of Hénon maps

“…This is known as the Pruning Front Conjecture. Mendonza proved that the Pruning Front Conjecture holds in an open set of the parameter space [Men13], and Ishii proved it for the Lozi family [Ish97]. For more detail on the theory of pruning fronts and its relationship with kneading theory, see [dCH02,dCH03].…”