We study the Holomorphic and Random Dynamics of some rank 2 free groups generated by two Hénon type maps. For these simply constructed examples we prove that the Fatou set is non-empty and that the stationary measures are supported on a compact set. With some further care this allows us to construct examples having no stationary measures. These examples illustrate the types of phenomena that may arise when studying holomorphic group actions on non-compact manifolds.

The results are part of an accepted manuscript. The Version of Record of that contribution will be published in Analysis and PDE in Developing Countries: Proceedings of the ISAAC-ICMAM Conference, 2024.