  There are two natural definitions of the Julia set for complex H\'enon maps:
the sets $J$ and $J^\star$. 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 $J^\star$, 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 $J = J^\star$ 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
$J^\star$. Substantially dissipative H\'enon maps admitting a dominated
splitting on the possibly larger set $J$ were recently studied in in [LP14].
