New Research on Hénon Map part2(Dynamical Systems)

1. The pruning front conjecture, wandering domains, and a classification of Hénon maps in the presence of strange attractors(arXiv)

Author : Jan P. Boroński, Sonja Štimac

Abstract : We study the topological dynamics of Hénon maps. For a parameter set generalizing the Benedicks-Carleson parameters (the Wang-Young parameter set) we obtain the following: The existence of wandering domains (answering a question of Lyubich, Martens, and van Strien); The pruning front conjecture (due to Cvitanović, Gunaratne, and Procacci); A kneading theory (realizing a conjecture by Benedicks and Carleson); A classification: two Hénon 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 classification result relies on a further development of the authors’ recent inverse limit description of Hénon attractors in terms of densely branching trees.

2.Hybrid dynamics of Hénon mappings (arXiv)

Author : Reimi Irokawa

Abstract : To study the meromorphic degeneration of complex dynamics, the theory of hybrid spaces, introduced by Favre, is known to be a strong tool. In this paper, we apply this theory to the dynamics of Hénon maps; for a family of Hénon maps {Ht}t∈D∗ parametrized by a unit punctured disk meromorphically degenerating at the origin, we show that as t→0, the family of the invariant measures {μt} ``weakly converges’’ to the measure on the Berkovich affine plane associated to the non-archimedean Hénon map determined by the family {Ht}t