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For the case of generic 4D symplectic maps with a mixed phase space weinvestigate the global organization of regular tori. For this we computeelliptic 1-tori of two coupled standard maps and display them in a 3Dphase-space slice. This visualizes how all regular 2-tori are organized arounda skeleton of elliptic 1-tori in the 4D phase space. The 1-tori occur in twotypes of one-parameter families: (a) Lyapunov families emanating fromelliptic-elliptic periodic orbits, which are observed to exist even far awayfrom them and beyond major resonance gaps, and (b) families originating fromrank-1 resonances. At resonance gaps of both types of families either (i)periodic orbits exist, similar to the Poincare-Birkhoff theorem for 2D maps, or(ii) the family may form large bends. In combination these results allow fordescribing the hierarchical structure of regular tori in the 4D phase spaceanalogously to the islands-around-islands hierarchy in 2D maps.

Global structure of regular tori in a generic 4D symplectic map