2017/06/01 by Lorenzo Guerini, Guerini, Lorenzo, Han Peters +1
Mathematics · Physics and Astronomy · #32H50 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1706.00220
openalex publication_date 2017/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
There are two natural definitions of the Julia set for complex Hénon maps: the sets J and J^⋆. Whether these two sets are always equal is one of the main open questions in the field. We prove equality when the map acts hyperbolically on the a priori smaller set J^⋆, under the additional hypothesis of substantial dissipativity. This result was claimed, without using the additional assumption, in the paper [For06], but the proof is incomplete. Our proof closely follows ideas from [For06], deviating at two points where substantial dissipativity is used. We show that J = J^⋆ also holds when hyperbolicity is replaced by one of two weaker conditions. The first is quasi-hyperbolicity, introduced in [BS02], a natural generalization of the one dimensional notion of semi-hyperbolicity. The second is the existence of a dominated splitting on J^⋆. Substantially dissipative Hénon maps admitting a dominated splitting on the possibly larger set J were recently studied in in [LP14].