Structural Instability of Semi-Siegel H\'enon maps

We show that the dynamics of sufficiently dissipative semi-Siegel complex H'enon maps with golden-mean rotation number is not $J$-stable in a very strong sense. By the work of Dujardin and Lyubich, this implies that the Newhouse phenomenon occurs for a dense $G_{\delta }$ set of parameters in this family. Another consequence is that the Julia sets of such maps are disconnected for a dense set of parameters.