## Institute for Mathematical Sciences

## Preprint ims16-01

** Remus Radu and Raluca Tanase**
*Semi-parabolic tools for hyperbolic Henon maps and continuity of Julia sets in C^2*

Abstract: We prove some new continuity results for the Julia sets $J$ and $J^{+}$ of the complex H\'enon 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 H\'enon 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 [RT] for the semi-parabolic
case to describe nearby perturbations. We show that for small nonzero $|t|$, the H\'enon 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 H\'enon maps is stable on $J$ and $J^{+}$ when $t$ is nonnegative.

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