Julia sets of complex Hénon maps

Abstract

There are two natural definitions of the Julia set for complex Hénon maps: the sets [Formula: see text] and [Formula: see text]. 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 [Formula: see text], under the additional hypothesis of substantial dissipativity. This result was claimed, without using the additional assumption, in [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457–464], but the proof is incomplete. Our proof closely follows ideas from [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457–464], deviating at two points, where substantial dissipativity is used. We show that [Formula: see text] also holds when hyperbolicity is replaced by one of the two weaker conditions. The first is quasi-hyperbolicity, introduced in [E. Bedford and J. Smillie, Polynomial diffeomorphisms of [Formula: see text]. VIII. Quasi-expansion. Amer. J. Math. 124(2) (2002) 221–271], a natural generalization of the one-dimensional notion of semi-hyperbolicity. The second is the existence of a dominated splitting on [Formula: see text]. Substantially dissipative, Hénon maps admitting a dominated splitting on the possibly larger set [Formula: see text] were recently studied in [M. Lyubich and H. Peters, Structure of partially hyperbolic hénon maps, ArXiv e-prints (2017)].