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

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.