Semi-parabolic tools for hyperbolic H\'enon maps and continuity of Julia sets in \mathbb{C}^{2}

We prove some new continuity results for the Julia sets J and J^{+} of the complex Hénon map H_{c,a}(x,y)=(x^{2}+c+ay, ax), where a and c are complex parameters. We look at the parameter space of dissipative Hénon maps which have a fixed point with one eigenvalue (1+t)\lambda , where \lambda is a root of unity and t is real and small in absolute value. These maps have a semi-parabolic fixed point when t is 0, and we use the techniques that we have developed in a prior work for the semi-parabolic case to describe nearby perturbations. We show that for small nonzero \vert t\vert, the Hénon map is hyperbolic and has connected Julia set. We prove that the Julia sets J and J^{+} depend continuously on the parameters as t\rightarrow 0, which is a two-dimensional analogue of radial convergence from one-dimensional dynamics. Moreover, we prove that this family of Hénon maps is stable on J and J^{+} when t is non-negative.