A structure theorem for semi-parabolic H\'enon maps

Consider the parameter space ${\mathcal{P}}_{\lambda }\subset {\mathbb{C}}^2$ of complex H'enon maps $$H_{c,a}(x,y)=(x^2+c+ay,ax),a\mathrm{\ne }0$$ which have a semi-parabolic fixed point with one eigenvalue $\lambda =e^{2\pi ip\mathrm{/}q}$. We give a characterization of those H'enon maps from the curve ${\mathcal{P}}_{\lambda }$ that are small perturbations of a quadratic polynomial $p$ with a parabolic fixed point of multiplier $\lambda$. We prove that there is an open disk of parameters in ${\mathcal{P}}_{\lambda }$ for which the semi-parabolic H'enon map has connected Julia set $J$ and is structurally stable on $J$ and $J^{+}$. The Julia set $J^{+}$ has a nice local description: inside a bidisk ${\mathbb{D}}_r\times {\mathbb{D}}_r$ it is a trivial fiber bundle over $J_p$, the Julia set of the polynomial $p$, with fibers biholomorphic to ${\mathbb{D}}_r$. The Julia set $J$ is homeomorphic to a quotiented solenoid.