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 \ensuremath{\alpha} direction, extending across both orientation-preserving and -reversing domains. There is also a secondary \ensuremath{\beta} 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 \ensuremath{\beta} direction. The \ensuremath{\alpha} direction is rich in new phenomena. The topology along \ensuremath{\alpha} is conjectured to be typical of bimodal maps.