Structure of the parameter space of the Hénon map

The parameter space of the H\'enon map is reported to contain a regular structure-parallel-to-structure sequence of shrimp-shaped robust isoperiodic domains. They appear densely concentrated on a neighborhood along a main direction, extending across both orientation-preserving and -reversing domains. There is also a secondary direction, roughly perpendicular to a very dense ``foliation of legs'' emanating from the isoperiodic domains. Familiar bifurcation phenomena observed in unimodal maps correspond to particular cuts along the direction. The direction is rich in new phenomena. The topology along is conjectured to be typical of bimodal maps.