The exit time decomposition of a set yields a description of the transport through the set as well as a visualization of the invariant structures inside it. We construct several sets computationally easier to deal with than the construction of resonances, based on the ordering properties for orbits of twist maps. Furthermore these sets can be constructed for four- and higher-dimensional twist mappings. For the four-dimensional case-using the example of Froeshle-we find "practically" invariant volumes surrounding elliptic fixed points. The boundaries of these regions are remarkably sharp; however, the regions are threaded by "tubes" of escaping orbits.