Structural Instability of Semi-Siegel H\'enon maps

TL;DR

This paper demonstrates that semi-Siegel Hénon maps with golden-mean rotation are dynamically unstable, leading to complex phenomena like the Newhouse phenomenon and disconnected Julia sets for many parameters.

Contribution

It establishes strong instability results for semi-Siegel Hénon maps, linking their dynamics to the occurrence of Newhouse phenomena and disconnected Julia sets.

Findings

Not $J$-stable in a strong sense for these maps.

Newhouse phenomenon occurs densely in parameter space.

Julia sets are disconnected for a dense set of parameters.

Abstract

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.