The pruning front conjecture, folding patterns and classification of H\'enon maps in the presence of strange attractors

TL;DR

This paper advances the understanding of Hénon maps by proving the pruning front conjecture, developing a kneading theory, and establishing a classification of strange attractors based on folding patterns and conjugacy.

Contribution

It proves the pruning front conjecture, develops a kneading theory, and provides a classification of Hénon attractors using folding patterns and inverse limit descriptions.

Findings

Proved the pruning front conjecture for Hénon maps.

Developed a kneading theory for classifying attractors.

Established a classification criterion based on folding patterns.

Abstract

We study the topological dynamics of H\'enon maps. For a parameter set generalizing the Benedicks-Carleson parameters (the Wang-Young parameter set) we obtain the following: The pruning front conjecture (due to Cvitanovi\'c); A kneading theory (realizing a conjecture by Benedicks and Carleson); A classification: two H\'enon maps are conjugate on their strange attractors if and only if their sets of kneading sequences coincide, if and only if their folding patterns coincide. The folding pattern is a single sequence of 0s and 1s, which allows to distinguish two nonconjugate H\'enon attractors in finitely many steps. The classification result relies on further development of the authors' recent inverse limit description of H\'enon attractors in terms of densely branching trees.